Single Degree Of Freedom Oscillator Research Articles

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.

Investigation of Soil Layers Stochasticity Effects on the Spatially Varying Seismic Response Spectra

The present study investigates the contribution of the stochasticity of a layered soil deposit on the seismic ground motion spatial variability via the random vibration theory. The seismic response evaluation is based on the 1-D shear wave propagation through random media. The soil layers stochasticity is introduced as spatial random fields of mass density and shear modulus of the soil deposit. The stochastic variation of these two parameters results in random natural frequency and damping ratio of an equivalent single degree of freedom oscillator modelling the soil deposit. Under the assumption of stationary Gaussian random process of the incident motion, the ground surface response spectra are obtained from the cross power spectral density of the total ground surface motion. The efficiency and the validity of the present approach are conducted via some applications. Numerical results showed that, the shape of the global coherency is controlled by that of the incident motion. The stochastic aspect of the soil layers causes disturbances in the surface motion and affects the coherency and the variability of the ground surface response spectra.

An improved replacement oscillator approach for soil-structure interaction analysis considering soft soils

This paper aims to improve the effectiveness of the replacement oscillator approach for soil-structure interaction (SSI) analysis of flexible-base structures on soft soil deposits. The replacement oscillator approach transforms a flexible-base single-degree-of-freedom (SDOF) structure into an equivalent fixed-base SDOF (EFSDOF) oscillator so that response spectra for fixed-base structures can be used directly for SSI systems. A sway-rocking SSI model is used as a baseline for assessment of the performance of EFSDOF oscillators. Both elastic and constant-ductility response spectra are studied under 20 horizontal ground motion records on soft soil profiles. The effects of frequency content of the ground motions and initial damping of the SSI systems are investigated. It is concluded that absolute acceleration spectra, instead of pseudo-acceleration spectra, should be used for EFSDOF oscillators in force-based design of SSI systems. It is also shown that using an EFSDOF oscillator is not appropriate for predicting the constant-ductility spectra when the initial damping ratio of the SSI system exceeds 10%. Based on the results of this study, a correction factor is suggested to improve the accuracy of the replacement oscillator approach for soft soil conditions.

Response of urban single and twin circular tunnels subjected to transversal ground seismic shaking

The response of tunnels subjected to ground seismic shaking is commonly evaluated, disregarding the existence of other structures in the near area. However, significant interaction phenomena may take place between the tunnel and existing structures in dense urban environments, altering the seismic response of the tunnel compared to the one predicted for ‘green field’ conditions. This study aims at investigating the effect of heavy buildings on the response of urban single or twin circular tunnels, when subjected to severe ground seismic shaking in the transversal direction. For this purpose, a numerical parametric study was conducted on representative tunnel-soil-buildings systems, employing the finite element code ABAQUS. Several configurations were examined, including single or twin circular tunnels, embedded in clayey soil deposits. The high-rise buildings were simulated as equivalent single degree of freedom oscillators (SDOFs), founded on rigid raft foundations. Salient parameters that affect the seismic response of tunnels, namely (i) the burial depth of the tunnels, (ii) the number of buildings-SDOFs and their position relative to the axis of the tunnel and (iii) the soil properties and response during ground shaking were investigated within this parametric study. The computed responses of the investigated configurations, in terms of (i) lining deformations, (ii) dynamic earth pressures and seismic shear stresses developed along the perimeter of the tunnels and (iii) dynamic lining forces, were compared with the cases where the buildings-SDOFs at the ground surface were neglected. The effect of buildings-SDOFs was quantified by means of response ratios μ, defined as the ratio of the maximum envelope value of the response parameter computed at critical lining section at the presence of the buildings-SDOFs, to the relevant value estimated for green-field conditions. The response ratios were plotted against the flexibility ratios, F, of the investigated soil-tunnel systems, highlighting the effects of the salient parameters mentioned above on the interaction phenomena. Generally, the consideration of the buildings-SDOFs at the ground surface resulted in an increase of the response of the tunnels, compared to the one predicted for ‘green field’ conditions, both in terms of tunnel deformations and dynamic earth pressures and soil dynamic shear stresses around the tunnel, as well as in terms of dynamic lining forces, with the effect being generally higher for shallow tunnels.

Discontinuous dynamics of a class of oscillators with strongly nonlinear asymmetric damping under a periodic excitation

Starting from the quarter car suspension system, the discontinuous dynamics of a general class of strongly nonlinear single degree of freedom oscillators are investigated using the flow switchability theory of the discontinuous dynamical systems. The characteristic of this oscillator is that they possess piecewise linear damping properties, which can be expressed in a general asymmetric form. More specifically, the viscous and constant damping properties appearing in the equation of motion depend on the velocity direction. Different domains and boundaries are defined according to the discontinuity. Based on above domains and boundaries, the analytical conditions for motion switchability at the velocity boundary in such oscillators are developed to understand the motion switching mechanism. To describe different motions in domains, the generic mappings and mapping structures are introduced. Based on the appropriate mapping structures, the periodic motions of such discontinuous systems are predicted analytically. Specified periodic and grazing motions for the quarter car model are given through the displacement, velocity and forces responses to illustrate the analytical criteria of complex motions. However, the periodic motions with switching for such nonlinear oscillators cannot be obtained from the traditional analysis, like the perturbation and harmonic balance method. Moreover, the present analysis can be extended to cover wider classes of dynamical systems, like mechanical oscillators with variable stiffness and damping properties.

Twin rotor damper for the damping of stochastically forced vibrations using a power-efficient control algorithm

The twin rotor damper (TRD), an active mass damper, uses the centrifugal forces of two eccentrically rotating control masses. In the continuous rotation mode, the preferred mode of operation, the two eccentric control masses rotate with a constant angular velocity about two parallel axes, creating, under further operational constraints, a harmonic control force in a single direction. In previous theoretical work, it was shown that this mode of operation is effective for the damping of large, harmonic vibrations of a single degree of freedom (SDOF) oscillator. In this paper, the SDOF oscillator is assumed to be affected by a stochastic excitation force and consequently responds with several frequencies. Therefore, the TRD must deviate from the continuous rotation mode to ensure the anti-phasing between the harmonic control force of the TRD and the velocity of the SDOF oscillator. It is found that the required deviation from the continuous rotation mode increases with lower vibration amplitude. Therefore, an operation of the TRD in the continuous rotation mode is no longer efficient below a specific vibration-amplitude threshold. To additionally dampen vibrations below this threshold, the TRD can switch to another, more energy-consuming mode of operation, the swinging mode in which both control masses oscillate about certain angular positions. A power-efficient control algorithm is presented which uses the continuous rotation mode for large vibrations and the swinging mode for small vibrations. To validate the control algorithm, numerical and experimental investigations are performed for a single degree of freedom oscillator under stochastic excitation. Using both modes of operation, it is shown that the control algorithm is effective for the cases of free and stochastically forced vibrations of arbitrary amplitude.

A modified hybrid Van der Pol–Duffing–Rayleigh oscillator for modelling the lateral walking force on a rigid floor

The paper proposes a single degree of freedom oscillator in order to accurately represent the lateral force acting on a rigid floor due to human walking. As a pedestrian produces itself the energy required to maintain its motion, it can be modelled as a self-sustained oscillator that is able to produce: (i) self-sustained motion; (ii) a lateral periodic force signal; and (iii) a stable limit cycle. The proposed oscillator is a modification of hybrid Van der Pol–Duffing–Rayleigh oscillator, by introducing an additional nonlinear hardening term. Stability analysis of the proposed oscillator has been performed by using the energy balance method and the Lindstedt–Poincare perturbation technique. Model parameters were identified from the experimental force signals of ten pedestrians using the least squares identification technique. The experimental and the model generated lateral forces show a good agreement.

Residual displacement demands of conventional and dual oscillators subjected to earthquake ground motions characteristic of the soft soils of Mexico City

This paper presents a statistical evaluation of residual displacement (RDs) demands in conventional and dual single-degree-of-freedom (SDOF) oscillators subjected to a set of 220 earthquake ground motions recorded at soft soil sites of Mexico City. A dual SDOF oscillator represents a highly-dissipating energy system (e.g. buckling-restrained braces, BRBs) acting in parallel with a conventional system (e.g. flexible moment-resisting frames). To provide a context, RDs were normalised with respect to the corresponding: a) maximum transient displacements, and b) elastic spectral displacement. The effects of post-yield stiffness ratio (as affected by P-Δ effects), normalised period of vibration with respect to the predominant period of the ground motion, the type of hysteretic response, maximum displacement ductility, lateral strength ratio, type of transition (from elastic-to-plastic response), and damping ratio on normalised RDs were examined for the conventional oscillators. In addition, the effects of the stiffness ratio, lateral strength ratio, displacement ductility, post-yielding stiffness ratio, and type of hysteretic response of the primary and secondary parts of dual SDOF oscillators on normalised RDs were also evaluated. From the results of this investigation, it was observed that for dual SDOF systems the amplitude of normalised RDs is small when the primary part remains elastic. On the contrary, if the primary part exhibits inelastic response, normalised RDs might increase significantly and the post-yielding stiffness ratio of the secondary part plays a key role for constraining them (i.e. while a positive value reduces RDs significantly; a negative value is highly detrimental). Also, it was found that the type of hysteretic response of the primary part of a dual system has a significant effect on normalised RDs (e.g. it was found that a primary part with self-centring capacity, acting in parallel with a highly dissipation system (e.g. BRBs) as secondary part, is very effective for diminishing RDs). A discussion section is also offered to highlight the new findings of this study and differences on normalised RD demands between soft and stiff soils.

The vibration transmissibility of a single degree of freedom oscillator with nonlinear fractional order damping

ABSTRACTMotivated by the theoretical analysis of the effects of nonlinear viscous damping on vibration isolation using the output frequency response function approach, the output frequency response function approach is employed to investigate the effects of the nonlinear fractional order damping on vibration isolation based on Volterra series in the frequency domain. First, the recursive algorithm which is proposed by Billings et al. is extended to deal with the system with fractional order terms. Then, the analytical relationships are established among the force transmissibility, nonlinear characteristic coefficients and fractional order parameters for the single degree of freedom oscillator. Consequently, the effects of the nonlinear system parameters on the force transmissibility are discussed in detail. The theoretical analysis reveals that the force transmissibility of the oscillator is suppressed due to the existence of the fractional order damping, but presents different effects on suppressing the force transmissibility of the oscillator over the frequency region by varying the fractional order parameters. Moreover, the fractional order parameters, which affect the force transmissibility, the bandwidth of the frequency region and the resonance frequency, can be used as designing parameters for vibration isolation systems. At last, numerical studies are presented to illustrate the theoretical results.

Closed-Form Steady-State Response Solution of the Twin Rotor Damper and Experimental Validation

In previous research, the twin rotor damper (TRD), an active mass damper, was presented including control algorithms for monofrequent vibrations. In a preferred mode of operation, the continuous rotation mode, two eccentric masses rotate in opposite directions about two parallel axes with a mostly constant angular velocity. The resulting control force is harmonic. Within this paper, the steady-state response of a single-degree-of-freedom (SDOF) oscillator subjected to a harmonic excitation force with and without the TRD is studied. A closed-form solution is presented and validated experimentally. It is shown that the TRD provides damping to the SDOF oscillator until a certain frequency ratio is reached. The provided damping is not only dependent on the design parameters of the TRD but also depends on the steady-state vibration amplitude. The solution serves as a powerful design tool for dimensioning the TRD. The analytical closed-form solution is applicable for other active mass dampers.

Energy harvesting based on magnetostriction, for low frequency excitations

Dynamics of an energy harvesting device based on Magnetostrictive material (MSM), Metglas, subjected to low frequency base excitation is studied. The model consists of a steel based cantilever beam laminated by Metglas as a MSM throughout the length. The cantilever beam is surrounded by a pickup coil on which electrical current is induced due to the magnetic field according to Faraday's law. The governing mechanical equation of the motion in conjunction with the equation describing the output electrical circuit are discretized and numerically integrated over the time. Unlike piezoelectric based energy harvesters, the proposed model offers low frequency energy harvesting. In the absence of base excitation, a free vibration problem subjected to initial condition is studied and the temporal response is determined. Due to the conversion of mechanical energy to electrical energy throughout the output circuit, the response resembles that of a damped single degree of freedom oscillator and the equivalent non-dimensional damping coefficient is determined. The steady state output power in terms of the excitation frequency is determined and the corresponding steady state current and voltage are presented. It is concluded that the non-dimensional damping coefficient exhibits Lorenzian response in terms of the load resistance indicating the multi-factorial dependency of the power on the governing parameters.

Alternating the twin rotor damper between two modes of operation to eliminate small vibrations

In previous research, the twin rotor damper (TRD), an active mass damper, was presented. In a preferred, energy-efficient mode of operation, the continuous rotation mode, two eccentric control masses rotate with constant angular velocities about two parallel axes in opposite directions. The resulting centrifugal forces combine into a directed harmonic control force that can be used for the control of structural vibrations. In another, more power-demanding mode of operation, the swinging mode, both control masses oscillate about certain angular positions. For both modes of operation, corresponding control algorithms have been developed. Furthermore, it has been found that the continuous rotation mode is not efficient for the damping of small vibrations, whereas the swinging mode can bring small vibrations to rest. In this paper, a control algorithm is presented, which uses the continuous rotation mode for large vibrations and the swinging mode for small vibrations. To validate the control algorithm, the TRD is numerically and experimentally applied on a single degree of freedom oscillator. The efficiency of the algorithm is studied for free vibrations.

Active vibration control of an oscillator with two translational degrees of freedom using centrifugal forces created by two eccentrically rotating masses

An active mass damper for vibration control of an oscillator with two translational degrees of freedom is presented along with the corresponding closed-loop control algorithm. The damper consists of two eccentrically rotating masses. In a preferred mode of operation, the masses rotate in opposite directions with a mostly constant angular velocity about a single axis. The resulting force is a harmonic control force. Its direction is determined by the relative angular position of the masses. In previous research, a similar control algorithm for vibration control of a single degree of freedom oscillator has been proven to be effective. In this paper, the control algorithm is augmented such that it can be applied to an oscillator with two translational degrees of freedom. Various state variables are introduced and a feedback control algorithm is developed. The presented algorithm ensures that the rotational motion of the masses is smooth and that the control force has the required orientation. The algorithm is verified experimentally with a test setup for the damping of free vibrations and numerically for stochastically forced vibrations. Finally, the device is compared with a conventional active mass damper. It is shown that the power demand and the energy consumption of the presented device are smaller than those of the conventional active mass damper.

On Response of a Single-Degree-of-Freedom Oscillator with Constant Hysteretic Damping Under Arbitrary Excitation

The simple constant hysteretic damping model is known to be non-causal although it is used often in diverse branches of engineering. In this paper the response of a single degree of freedom oscillator having linear hysteretic damping under arbitrary force excitation has been studied after deriving the impulse response function of the system. Some shortcomings of the results available in literature have been pointed out. It has been shown that the damping model can be practically used for calculating the response of a physical system when the damping is small and the force has small duration.

An AutomaticP‐Phase Arrival‐Time Picker

Presented is a new approach for picking P ‐phase arrival time in single‐component acceleration or broadband velocity records without requiring detection interval or threshold settings. The algorithm P PHASE P ICKER transforms the signal into a response domain of a single‐degree‐of‐freedom (SDOF) oscillator with viscous damping and then tracks the rate of change of dissipated damping energy to pick P ‐wave phases. The SDOF oscillator has a short natural period and a correspondingly high resonant frequency, which is higher than most frequencies in a seismic wave. It also has a high damping ratio (60% of critical). At this damping level, the frequency response approaches the Butterworth maximally flat magnitude filter, and phase angles are preserved. The relative input energy imparted to the oscillator by the input signal is converted to elastic strain energy and then dissipated by the damping element as damping energy. The damping energy yields a smooth envelope over time; it is zero in the beginning of the signal, zero or near zero before the P ‐phase arrival, and builds up rapidly with the P wave. Because the damping energy function changes considerably at the onset of the P wave, it is used as a metric to track and pick the P ‐phase arrival time. The P PHASE P ICKER detects P ‐phase onset using the histogram method. Its performance is compared with picking techniques using short‐term‐average to long‐term‐average ratio, and a picking method that finds the first P ‐phase arrival time using the Akaike information criterion. A large set of records with various intensities and signal‐to‐noise ratios is used for testing the P PHASE P ICKER , and it is demonstrated that P PHASE P ICKER is able to more accurately pick the onset of genuine signals against the background noise and to correctly distinguish between whether the first arrival is a P wave (emergent or impulsive) or whether the signal is from a faulty sensor. Online Material: MATLAB script for P ‐phase arrival time picking.

Nonlinear dynamic response of a base-excited SDOF oscillator with double-side unilateral constraints

The aim of this paper was to study the dynamic response of a single-degree-of-freedom (SDOF) oscillator exited by a base acceleration and constrained by two unilateral constraints (bumpers). The coupled equations of motion are formulated in dimensional and dimensionless form for the case in which both SDOF oscillator and bumpers exhibit a nonlinear behavior. Two possible states characterize the system’s response: flight, when the mass of the oscillator is not in contact with the bumper, and contact, when the mass touches the bumper. When the system is in the flight state, the independent time evolution of the bumpers’ response is also considered in equations of motion. In the numerical investigations, a viscoelastic SDOF oscillator and viscoelastic perfectly plastic bumpers subjected to a harmonic base excitation are considered. First, the transient versus steady-state dynamic response is considered, and afterward the only steady-state dynamic response is studied by means of pseudo-resonance curves of maximum absolute acceleration and relative displacement excursion of the SDOF oscillator. Furthermore, the continuation technique is applied in some cases in order to highlight in the system’s dynamic response hysteresis ranges, jumps between multi-periodic orbits, and super-harmonics. The dynamic analysis with and without bumpers is compared to observe how unilateral constraints modify the dynamic response of the SDOF oscillator with respect to the absence of bumpers and when their presence may be beneficial with respect to response mitigation.

A constant phase approach for the frequency response of stochastic linear oscillators

When studying a mechanical structure, evaluation of its frequency response function (FRF) over a given frequency range is one of the main interests. Computational cost aside, evaluating FRFs presents no methodological difficulty in the deterministic case. Doing this when the model includes some uncertain parameters may however be more difficult as multimodality and discontinuity can arise around resonances. Indeed, even for a single degree of freedom system, it can be shown that usual methods of the probabilistic frame such as generalized Polynomial Chaos may fail to properly describe the probability density function of the response amplitude. This study proposes another approach which involves a shift in the usual quantities used to draw FRFs. Instead of computing the stochastic response for a given excitation frequency, this work adopts a constant response phase point of view. For each phase value of the oscillator response, the uncertainty over some parameters is propagated to the corresponding uncertain amplitudes and excitation frequencies. This provides much smoother variations of the involved quantities which are much easier to describe using a simple Polynomial Chaos approach. Both analytical and numerical results will be exposed for a single degree of freedom oscillator whose stiffness follows a uniform law.

Seismic Assessment of Reinforced Concrete Frameworks through Advanced Pushover Analysis and Nonlinear Response of a SDOF Oscillator

This paper presents an integrated system for advanced structural analysis and seismic performance evaluation of 2D reinforced concrete frameworks. The advanced non-linear inelastic static analysis employed herein uses the accuracy of the fibre elements approach for large deflection inelastic frame analysis and address its efficiency and modelling shortcomings both to element level, through the use of only one element to model each physical member of the frame, and to cross-sectional level through the use of path integral approach for numerical integration of the cross-sectional nonlinear characteristics. Evaluation of the seismic performance is achieved with an approach that uses nonlinear time-history analysis (NTHA) of a single degree-of-freedom (SDOF) oscillator. The inelastic ductility and displacement demand is determined directly from accelerograms, without graphical or numerical approximations. Several computational examples are given to validate the effectiveness of the proposed method, the reliability and time saving of the code. The influence of the accuracy of the second-order nonlinear inelastic analysis methods involved in the modelling of the selected RC frame in conjunction with the distributions of the pushover lateral loads is pointed out. The proposed procedure is developed in the framework of Eurocode 8 design methodology, as reliable tool ready to be implemented into everyday design practice for advanced analysis, pushover analysis, and seismic performance evaluation of RC frame structures.

Universal response spectrum procedure for predicting walking-induced floor vibration

Floor vibrations caused by people walking are an important serviceability problem both for human occupants and vibration-sensitive equipment. Present design methodologies available for prediction of vibration response due to footfall loading are complex and suffer from division between low and high frequency floors. In order to simplify the design process and to avoid the problem of floor classification, this paper presents a methodology for predicting vibration response metrics due to pedestrian footfalls for any floor type having natural frequency in the range 1–20Hz.Using a response spectrum approach, a database of 852 weight-normalised vertical ground reaction force (GRF) time histories recorded for more than 60 individuals walking on an instrumented treadmill was used to calculate response metrics. Chosen metrics were peak values of 1s peak root-mean-square (RMS) acceleration and peak envelope one-third octave velocities. These were evaluated by weight-normalising the GRFs and applying to unit-mass single degree of freedom oscillators having natural frequencies in the range 1–20Hz and damping ratios in the range 0.5–5%. Moreover, to account for effect of mode shape and duration of crossing (i.e. duration of dynamic loading), the recorded GRFs were applied for three most typical mode shapes and floor spans from 5m to 40m.The resulting peak values as functions of frequency i.e. spectra are condensed to statistical representations for chosen probability of being exceeded over a wide range of applications. RMS (acceleration) spectra show strong peaks corresponding to the first harmonic of pacing rate followed by clear minima at approximately 3.5Hz, a second much smaller peak corresponding to the second harmonic and a steady decline with increasing frequency beginning around 5Hz. One-third octave spectra show asymptotic trends with frequency, span and damping.A comprehensive validation exercise focusing on the acceleration RMS spectra was based on a representative range of floor samples for which modal properties had been identified and walking response studied during experimental campaigns of vibration serviceability evaluation. Due to the statistical approach an exact validation would not be possible, hence measured peak RMS values were matched to distributions for the equivalent idealised structure. In the vast majority of cases the measured values, intended to represent worst-case conditions, fitted the upper decile of the corresponding simulated spectra indicating consistency with the proposed approach.