Simple Single Degree Of Freedom Research Articles

Mathematical Modeling of the Floating Sleeper Phenomenon Supported by Field Measurements

This article aims to provide an accurate mathematical model with the minimum number of degrees of freedom for describing the floating sleeper phenomenon. This was accomplished using mathematical modeling supported by extensive field measurements of the railway track. Although the observed phenomenon is very complex, the simplified single degree of freedom (SDOF) mathematical model proved accurate enough for its characterization. The progression of the deterioration of the railway track was successfully correlated to changes in the maximal dynamic factor for different types of pulse loading. The results of the presented study might enable the enhanced construction and maintenance of railroads, particularly in karst areas.

A compatible uniaxial Bouc–Wen model for accurate estimation of residual deformation of seismically isolated structures

Bouc–Wen (BW) model is commonly used to model lead rubber bearings (LRBs) in seismically isolated buildings. However, the original BW model violates the Drucker’s or Ilyushin’s postulate of plasticity by displaying the displacement drift, force relaxation, and non-closure of hysteretic loops. These deficiencies can significantly impact the seismic performance of the isolated structures, leading to unrealistic evaluation of residual deformation concentrated in the isolation layer. To address these issues, a step-by-step amendment algorithm is proposed to develop a compatible uniaxial BW (CBW) model that conforms to the basic physical rules. The proposed algorithm involves adding a reinforcement factor to the short-path reloading branches, defining reversal and reloading points to identify reloading behavior. Furthermore, it develops an adaptive limit to distinguish between the short-path and full-path reloading branches. The proposed algorithm was validated using various short unloading–reloading scenarios. Subsequently, a simplified single degree of freedom (SDOF) model representing a seismically isolated building was defined. Incremental dynamic analysis was performed to evaluate the seismic responses of the isolated building designed with the original BW, CBW, and bilinear models. Results indicate that although the peak displacement and seismic force of the isolation layer induced by the three models produce small deviations, considerable errors in residual displacement are observed. Compared with the original BW model, the bilinear model produces unrealistic sharp yielding transition and overestimates the residual displacement. However, it can reduce displacement drift and force relaxation. By contrast, the proposed CBW with the improved algorithm can effectively correct the defects of the original BW and bilinear models. The CBW model can be used to accurately estimate the residual deformation of the seismically isolated buildings.

HORIZONTAL ACCELERATION OF FLOOR-MOUNTED NONSTRUCTURAL COMPONENTS UNDER ROCKING MOTION

Floor-mounted nonstructural components (NSCs) that are attached through various types of angles are susceptible to seismic damage as they experience severe rocking motion resulting from their slender geometric configuration. However, current design codes do not reflect such complex dynamic behavior and prescribe design factors based on simplified single-degree-of-freedom (SDOF) models, which might cause an under-estimation of seismic design forces. Therefore, to properly control their seismic damage, the complex rocking dynamic behavior needs to be thoroughly investigated. This paper analyzed the rocking behavior of the floor-mounted NSCs and their effects on the horizontal acceleration response. Floor-mounted NSCs and their mounting attachments were idealized as rigid blocks restrained by linear vertical springs at the corner of the block. The nonlinear equations for the rocking behavior of rigid block were developed, and the analysis was conducted based on the fourth-order Runge-Kutta integration method. First, it is shown that the resonance period of rocking NSCs is mainly affected by the NSC slenderness and the location of the center of gravity of the component. The horizontal acceleration tends to increase as the slenderness and the height of the center of gravity of NSCs increase. Based on the observations, some useful design recommendations were suggested for designing attachments of floor-mounted NSCs.

Harmonizing Seismic Performance via Risk Targeted Spectra: State of the art, dependencies, and implementation proposals

AbstractDesign spectra are commonly defined for a constant value of hazard associated to an “ultimate” limit state (LS) of reference, typically 10% in 50 years. Given a specific structure, this approach results in different limit‐state exceedance risk levels even for sites characterized by the same design peak ground acceleration (PGA), mainly because of differences in hazard curve shape/slope. Instead, Risk Targeted design maps (first applied in ASCE7‐10) suggest the application of suitable spectra adjustment factors, the so‐called risk coefficients, in order to ensure a uniform collapse risk across different sites and buildings. Making use of simplified single‐degree‐of‐freedom (SDoF) structures for several configurations of vibration period and ductility, we test the effectiveness of the adjustment factors computed under different assumptions in matching a specific target risk or, at least, in harmonizing the risk of multiple buildings at different sites with respect to different LSs. Although risk matching is shown to be only theoretically possible and unachievable in practice, we claim that harmonization remains a viable target, and we offer insights for possible future adoption of Risk Targeted Spectra in the European code provisions.

Dynamic response of a Single-Degree-of-Freedom system containing Phase Transforming Cellular Materials

Repeatable superelastic behavior is a much sought after attribute in the field of earthquake engineering. Phase Transforming Cellular Materials (PXCMs) are architected materials that exhibit superelastic force–deformation behavior. PXCMs can be designed to exhibit nearly perfect elasto-plastic behavior, meaning they can undergo large deformations at nearly constant force. Moreover, PXCMs can return to their original configuration without permanent deformation on removal of the external load. The viability of integrating PXCMs in Seismic Force Resisting Systems (SFRS) of structures was investigated through dynamic testing of a simple Single-Degree-of-Freedom (SDOF) PXCM system. The response of the SDOF PXCM system was also compared with the response of systems constructed using traditional materials such as reinforced concrete. The tests and comparisons showed that 1) PXCMs are capable of exhibiting nonlinear behavior without accumulating damage when subjected to dynamic loads, and 2) PXCM systems are likely to displace less than similar systems constructed using traditional materials. Hence, it was concluded that PXCMs can supplement existing building materials to enhance the overall seismic performance of a structure.

Effect of Viscous Damping Models on Displacement Ductility Demands for SDOF Systems

The viscous damping is a highly idealized but mathematically convenient way of representing the energy dissipation mechanisms not related to the hysteretic response. In this study, the tangent-stiffness proportional Rayleigh damping (TSPRD) appropriate for inelastic hysteretic structures outfitted with the supplemental damping devices is utilized in inelastic dynamic analysis of simple single-degree-of-freedom (SDOF) structures. A numerical procedure of dynamic equilibrium equation for SDOF systems with TSPRD model is derived for development of the general numerical solution scheme. By specifying various combinations of tangent-stiffness and mass proportional damping terms in TSPRD model, the influence of viscous damping models on displacement ductility demands is investigated. The results indicate that the difference in ductility demands for various damping models is considerable, depending on the periods of vibration, relative lateral strength, hysteretic models, stiffness deterioration and initial damping ratios. The constant-strength displacement ductility spectra, by considering both tangent-stiffness proportional and mass proportional damping, are developed in terms of site conditions, hysteretic rules and initial damping ratios.

Blast Response of Single-Degree-of-Freedom Systems Including Fluid-Structure Interaction

AbstractThe state of the practice for the blast-resistant design of structures uses a simplified single-degree-of-freedom (SDOF) analysis and estimates the blast loads assuming reflection from an u...

Reduced-Order Modeling and Experimental Studies of Bilaterally Coupled Fluid–Structure Interaction in Single-Degree-of-Freedom Flapping Wings

AbstractFlapping wings deform under both aerodynamic and inertial forces. However, many flapping wing fluid–structure interaction (FSI) models require significant computational resources which limit their effectiveness for high-dimensional parametric studies. Here, we present a simple bilaterally coupled FSI model for a wing subject to single-degree-of-freedom (SDOF) flapping. The model is reduced-order and can be solved several orders of magnitude faster than direct computational methods. To verify the model experimentally, we construct a SDOF rotation stage and measure basal strain of a flapping wing in-air and in-vacuum. Overall, the derived model estimates wing strain with good accuracy. In-vacuum, the wing has a large 3ω response when flapping at approximately one-third of its natural frequency due to a superharmonic resonance, where the superharmonic occurs due to the interaction of inertial forces and time-varying centrifugal softening. In-air, this 3ω response is attenuated significantly as a result of aerodynamic damping, whereas the primary ω response is increased due to aerodynamic loading. These results highlight the importance of (1) bilateral coupling between the fluid and structure, since unilaterally coupled approaches do not adequately describe deformation-induced aerodynamic damping and (2) time-varying stiffness, which generates superharmonics of the flapping frequency in the wing’s dynamic response. The simple SDOF model and experimental study presented in this work demonstrate the potential for a reduced-order FSI model that considers both bilateral fluid–structure coupling and realistic multi-degrees-of-freedom flapping kinematics moving forward.

Seismic Response of a Structure Equipped with an External Viscous Damping System

The aim of this research is to evaluate the effectiveness of a seismic retrofit technique that involves the introduction of energy dissipation devices properly connected to an existing structure through a system of cables and levers, which are employed to amplify total or inter-story drift at device end. One of the main topics related to the introduction of energy dissipation devices, lies in the choice of their optimal setting within the structure to maximize the effectiveness without producing functionality limitations. The achievement of these objectives is, therefore, linked, regardless of the type adopted, to the amount of energy dissipated in each cycle, directly proportional to the displacement magnitude to which the device is subject. Many configurations proposed in the literature and currently adopted in professional practice provide additional dissipation systems variously connected to braces installed inside the structural frame and, therefore, able to exploit the inter-story drift produced by seismic input. The proposed system exploits top displacements of the structure with respect to the foundation level, transferred to the device through a system of cables properly configured and amplified with leverage. This paper represents the first step of the research, in which simple single degree of freedom (SDOF) or two degrees of freedom (2-DOF) models are taken into account to evaluate the effects of the introduction of the proposed system in terms of reducing the seismic demand on the structure, proceeding to a parametric analysis to obtain initial indications for the design of the system in relation to the geometric and inertial characteristics of the original structure.

Prediction of blast pressure-duration capacity of monolithic Thermally Tempered Glass panes

Most standards and documents for blast-resistant design of glazing systems are based on simplified Single Degree Of Freedom (SDOF) models and triangular blast pressure time histories with no negative phase. Previous experimental studies have clearly indicated that pressure-duration or pressure-impulse design curves generated from such SDOF models may be overconservative, especially during impulsive blast scenarios. In this study, pressure-duration capacity curves for various monolithic Thermally Tempered Glass (TTG) panes are derived using Finite Element (FE) models. Moreover, the imposed blast pressure time histories include both positive and negative phases, which are deemed to be more realistic than considering the positive phase alone. Previous blast tests have shown that TTG panes can experience significant in-plane strain before fracture and exceeding the cracking strength does not necessarily mean fracture and disintegration of the pane. Accordingly, in the present study, glass fracture is judged based on the maximum principal strain rather than the stress criterion which has been widely used in earlier studies. The accuracy of the FE modeling technique and the abovementioned fracture criterion is verified using these blast test results on TTG panes. Pressure-duration curves of predicted capacity are obtained for TTG panes with different aspect ratios (1, 1.25, 1.5, 1.75, 2), different thicknesses (8 mm, 10 mm, 12 mm, 16 mm) and different widths (600 mm, 900 mm, 1,200 mm). Similar to current available standards, four edges of the adopted panes are restrained in the out-of-plane direction, but without in-plane or rotational restraints. The obtained pressure-duration capacity curves are compared with those provided by UFC 3-340-02. This shows that UFC's pressure-duration capacity curves are overconservative in both the impulsive and quasi-static regions.

Mathematical modeling of Electric vehicles - A survey

The paper presents a survey of the existing mathematical models of electric vehicles followed by state of the art approaches on modeling hybridization of the energy sources. The mathematical representation of the electric vehicle (EV) ranging from the simple single Degree of Freedom (DOF) models to complex multi-body dynamic models is discussed in detail. Reduced dynamic models applicable to various control design diligence are discussed which facilitates the selection of the optimal model for design. In addition to vehicle dynamics, the paper consolidates dynamic models of the different components of an electric vehicle including the transmission, brake, battery, wheel and tire dynamics. Comparative analysis of different versions of the models for each component is also presented, focusing on their application in controller design. The paper thus acts as a guide for any EV control design requirement, providing optimal models for particular applications.

Reliability of Field Experiments, Analytical Methods and Pedestrian’s Perception Scales for the Vibration Serviceability Assessment of an In-Service Glass Walkway

The vibration performance of pedestrian structures attracts the attention of several studies, especially with respect to unfavorable operational conditions or possible damage scenarios. Given a pedestrian system, specific vibration comfort levels must be satisfied in addition to basic safety requirements, depending on the class of use, the structural typology and the materials. To this aim, guideline documents of the literature offer simplified single-degree-of-freedom (SDOF) approaches to estimate the maximum expected vibrations and to verify the required comfort limits. Most of these documents, however, are specifically calibrated for specific scenarios/structural typologies. Dedicated methods of design and analysis, in this regard, may be required for structural glass pedestrian systems, due to their intrinsic features (small thickness-to-size ratios, high flexibility, type and number of supports, live-to-dead load ratios, use of materials that are susceptible to mechanical degradation with time/temperature/humidity, etc.). Careful consideration could be then needed not only at the design stage, but also during the service life of a given glass walkway. In this paper, the dynamic performance of an in-service glass walkway is taken into account and explored via field vibration experiments. A set of walking configurations of technical interest is considered, involving 20 volunteers and several movement features. The vibration comfort of the structure is then assessed based on experimental estimates and existing guideline documents. The intrinsic uncertainties and limits of simplified approaches of literature are discussed, with respect to the performance of the examined glass walkway. In conclusion, the test predictions are also used to derive “perception index” data and scales that could support a reliable vibration comfort assessment of in-service pedestrian glass structures.

Blast response of RC slabs with externally bonded reinforcement: Experimental and analytical verification

The present paper provides an analysis of the efficiency of externally bonded reinforcement (EBR) on reinforced concrete (RC) slabs under blast loads. Five simply supported slabs with a span of 2 m are tested under explosive charge. One of the slabs is used as a reference specimen and the remaining slabs were strengthened with different ratios of carbon fiber reinforced polymer (CFRP). An analytical analysis is carried out using the simplified single-degree-of-freedom (SDOF) approach to predict the maximum deflection at midspan. Digital image correlation (DIC) is used to measure the maximum deflection at the midspan of the slab and the strain distribution in the concrete and the EBR. Given the challenge to combine these displacement field measurements with blast, an explosive driven shock tube method has been adopted in this study. The results indicate that CFRP as EBR increases significantly the flexural capacity and the stiffness of RC slabs under blast loads. The impact of the blast wave on the RC slabs generates high strains in concrete, steel reinforcement and CFRP strips. Good correspondence in the prediction of the maximum deflection between the experimental and the analytical results, is obtained, showing that analytical analysis by means of the simplified SDOF approach leads to a reliable prediction.

Neural Network Based Vibration Control of Seismically Excited Civil Structures

This study proposes a neural network based vibration control system designed to attenuate structural vibrations induced by an earthquake. Classical feedback control algorithms are susceptible to parameter changes. For structures with uncertain parameters they can even cause instability problems. The proposed neural network based control system can identify the structural properties of the system and avoids the above mentioned problems. In the present study it is assumed that a full state of the structure is known, which means the at each floor horizontal displacements and rotations about the vertical axis are measured. Additionally, it is assumed the acceleration signal coming from the earthquake is also available. The proposed neural control strategy is compared with the classical linear quadratic regulator (LQR) not only in terms of displacement responses, but also required control forces. Moreover, the influence of different weighting matrices on performance of the proposed control strategy has been presented.The effectiveness of the neuro-controller has been demonstrated on two numerical examples: a simple single degree of freedom (DOF) structure and a multi-DOF structure representing a twelve story building. Both structures under consideration have been excited with El Centro acceleration signal. The results of numerical simulations on the SDOF system indicate that using neuro-controller it would be possible to obtain smaller amplitudes as compared with the LQ regulator, but it would require higher control effort.

Vibration control of a membrane antenna structure using cable actuators

This paper presents a control strategy of using cable actuators to control the vibration of a membrane antenna structure. The tension cables of the membrane antenna structure are used as actuators, the vibration of the structure can be suppressed by controlling the tension force in the cable actuators. First, the antenna structure with cable actuators is discretized by the finite element method (FEM), and governing equation of the whole structure is established. Then, a controller is designed based on the Lyapunov's stability theory, and the mechanism of this controller is studied through a simple single-degree-of-freedom (SDOF) system. The optimal placement of the cable actuators is also studied numerically in this paper. Simulation results indicate that vibration of the membrane antenna structure can be suppressed effectively by the cable actuators, and optimally placed cable actuators can produce better control effect with smaller control input.

Investigation on bit stick-slip vibration with random friction coefficients

Torsional vibration of a drillstring, bit stick-slip in particular, is investigated with a simplified single degree of freedom (DOF) model in this paper. The friction in the bit-rock interaction is modeled as random, and the motions in stick and slip stages are treated separately. In the stick stage, it is treated as in static equilibrium. A single point or a range of points, at which the stick stage is completed and the slip stage is started, are determined depending on deterministic or random static friction coefficient assumed respectively. In the slip stage, the response at a time instant becomes a spreading area centered around the means due to the randomness of the kinetic friction. By assuming the random kinetic friction as White noise, the probabilistic distribution of the responses with displacement and velocity is calculated with the numerical path integration (PI) technique. For verification purpose, Monte Carlo (MC) simulation is also conducted. Also due to the randomness in friction, the time instants leaving the slip stage, or entering the stick stage, become random. Comparison between PI and MC shows that results from the two methods are in good agreement. In addition, parametric studies on damping, rotary speed, weight on bit, drillstring length and different combination of pipe and collar are conducted for both deterministic and random cases.

Direct displacement-based design accuracy prediction for single-column RC bridge bents

In the last decade, displacement-based (DB) methods have become established design procedures for reinforced concrete (RC) structures. They use strain and displacement measures as seismic performance control parameters. As for other simplified seismic design methods, it is of great interest to prove if they are usually conservative in respect to more refined, nonlinear, time history analyses, and can estimate design parameters with acceptable accuracy. In this paper, the current Direct Displacement-Based Design (DDBD) procedure is evaluated for designing simple single degree of freedom (SDOF) systems with specific reference to simply supported RC bridge piers. Using different formulations proposed in literature for the equivalent viscous damping and spectrum reduction factor, a parametric study is carried out on a comprehensive set of SDOF systems, and an average error chart of the method is derived allowing prediction of the expected error for an ample range of design cases. Following the chart, it can be observed that, for the design of actual RC bridge piers, underestimation errors of the DDBD method are very low, while the overestimation range of the simplified displacement-based procedure is strongly dependent on design ductility.

Identification of Random Excitation Fields From Vibratory Responses With Application to Multisupported Tubes Excited by Flow Turbulence

In this paper, we address the identification of the spectral and spatial features of random flow excitations for multisupported tubular components such as steam generator tubes and nuclear fuel rods. In the proposed work, source identification is performed from a set of measured vibratory responses, in the following manner: (1) The modal response spectra and modeshape amplitudes at the measurement locations are first extracted through a blind decomposition of the physical response matrix, using the second order blind identification (SOBI) method; (2) the continuous modeshapes are interpolated from the identified values at the measurement locations; (3) the system modal parameters are identified from the modal responses using a simple single degree of freedom (SDOF) fitting technique; (4) inversion from the modal response spectra is performed for the identification of the modal excitation spectra; (5) finally, an equivalent physical excitation spectrum as well as the flow velocity profiles are estimated. The proposed approach is illustrated with identification results based on realistic numerical simulations of a multisupported tube under linear support conditions.

Dynamic Response of a RC Frame under Column Removal

AbstractUnder a collapse scenario, the sudden loss of support causes a dynamic response that can amplify the internal forces in the members and lead to greater damage. This study considered a one-quarter scale, 2-bay, 2-story RC frame that was axially restrained at the beam locations. The frame was loaded with dead weights, and the support under the center column was kicked out to initiate the dynamic test. The results from four dynamic tests of a reinforced concrete frame with discontinuous reinforcement under various levels of applied load are presented. The fourth drop, with a load corresponding to 42% of the design (1.2×dead load+0.5×live load), did result in a catenary action range of response. A simple single degree of freedom (SDOF) analysis showed that there is a snap-through effect, i.e., a very fine tipping point at which the structure is pushed past the compressive arch or flexural range of response into the catenary action range of response. At this point, the loads in the structure increase s...

A Theoretical and Experimental Investigation on Free Vibration Vehavior of a Cantilever Beam with a Breathing Crack

In this paper the free nonlinear vibration behavior of a cracked cantilever beam is investigated both theoretically and experimentally. For simplicity, the dynamic behavior of a cracked beam vibrating at its first mode is simulated using a simple single degree of freedom lumped parameter system. The time varying stiffness is modeled using a harmonic function. The governing equation of motion is solved by a perturbation method - the method of Multiple Scales. Results show that the correction term that is added to the main part of the response reflects the effect of breathing crack on the vibration response. Moreover, this part of response consists of the superharmonic components of the spectrum which is due to the system's intrinsic nonlinearity. Using this method an analytical relation is established between the system characteristics and the crack parameters in one hand and the nonlinear characteristics of system response on the other hand. Results have been validated by the experimental tests and a numerical method.