Single Degree Of Freedom Model Research Articles

Dynamic behaviors of unreinforced and spray polyurea retrofitted brick masonry infill walls under blast loads: Shock tube test and analyses

Unreinforced masonry infill walls (MIWs) are prone to crack, fragment, and even collapse under blast loads, threatening the safety of building structures, as well as the inside occupants and equipment. Spray polyurea (SPUA) has been proven to be an excellent retrofitted material for MIWs to improve their blast resistance. Aiming to study the dynamic behaviors of brick MIWs under far-field range blast loads, the successive blast tests on seven unreinforced and SPUA-retrofitted brick MIWs with different thicknesses and boundary constraints were first carried out based on the developed compressed air-driven large cross-section (3m×3 m) shock tube. It was indicated that the blast resistance of MIWs significantly improved by increasing wall thickness and strengthening boundary constraint; SPUA coating can improve the blast resistance of MIWs and effectively suppress the debris generation and splash. Then, the equivalent single degree of freedom model based theoretical calculation and the simplified micro-model based numerical simulation methods previously established by authors were briefly introduced. Additionally, the dynamic behaviors of seven examined MIWs were further predicted and compared with test results, which verified the applicability of theoretical calculation and numerical simulation methods. Finally, the improved damage and failure evaluation criteria of unreinforced and SPUA-retrofitted MIWs were given, and the charge weight and standoff distance diagrams of typical MIWs were proposed based on the validated theoretical and numerical methods as well as the improved criteria, which would provide a helpful reference for blast-resistant evaluation and retrofitted design of brick MIWs.

Experimental study on vibration control performance of TMD-STF damper

A tuned mass damper (TMD) using shear thickening fluid (STF) as an energy dissipation medium is designed and manufactured. In addition, a TMD incorporating silicone oil (TMD-Si49) is fabricated to compare and analyze the vibration control performance of TMD-STF. Free vibration, forced vibration, and random excitation tests are conducted on a single degree of freedom (SDOF) structure model equipped with TMD-STF and TMD-Si49, respectively. The displacement and acceleration response of the SDOF model are recorded to analyze the vibration control performance of TMD-STF and TMD-Si49 under different load conditions. The results of the free vibration test show that TMD-STF affects the coupling damping ratio of the test model, exhibiting time-varying damping characteristics, but does not impact the coupling frequency ratio. In the forced vibration test, TMD-STF effectively expands the effective frequency range of vibration control performance by approximately 17.0% compared to TMD-Si49. The random excitation test demonstrates that TMD-STF exhibits significantly improved control of acceleration response rather than displacement control. The research findings on TMD-STF provide novel insights for vibration control in engineering structures.

Noise analysis of interior structure under powertrain excitation

To accurately identify the structural noise problem caused by vibration in the cockpit, an experimental study was conducted on a certain type of SUV under uniform acceleration conditions. Based on the planning of the multi-body system dynamics model, a transfer path model with secondary contributions is constructed, and the load is identified using the single degree of freedom model and multi-frequency bandwidth model under the OPAX method based on this path. The results indicate that the second-order vibration of the powertrain is transmitted to the entire suspension system through the X direction of the right suspension. The second-order vibration of the suspension system is then transmitted to the Z direction of the right rear bolt of the subframe, causing the entire subframe vibration and ultimately reaching the target point. This structural path has the greatest impact on the interior noise of the vehicle. This path research method can more accurately determine the main excitation points that cause interior noise problems on a certain path and better understand the overall energy transfer characteristics along the transmission path.

Progressive collapse analysis of steel frames via a nonlinear SDOF model

AbstractProgressive collapse is a dynamic loading incident triggered by the sudden loss of critical elements. Design codes provide simplified rules to consider the dynamic effects via the use of Dynamic Increased Factors in order to avoid complex and time‐consuming nonlinear time history analyses. In this paper, a nonlinear single degree of freedom (SDOF) system is presented that captures the dynamic response of steel frames subjected to a column removal scenario. The model takes into account inertia, damping effects and assumes an elastoplastic nonlinear resistance for the system. A closed‐form analytical solution for the dynamic equation of motion is derived and then is applied to a frame to simulate the response under a column removal scenario. A nonlinear finite element model developed in OpenSees was used to verify the accuracy of the analytical model. The results showed that the nonlinear response of the frame for the selected hazard event can be well captured with the proposed model.

Human-structure dynamic interactions: Identification of two-degrees-of-freedom walking human model

Single-degree-of-freedom (SDOF) models are widely used as a first approximation to simulate dynamic interactions of a walking human body with a vertically vibrating structure. However, the parameters identified in the literature for such SDOF models are distinctly different for different structures and even different modes of a structure. This study hypothesises that these differences can be attributed to different modes of a human body being dominant when subjected to different vibration frequencies. A 2DOF mass-spring-damper model with two modes of vibration is proposed to simulate a walking human dynamics. The 2DOF model parameters were identified using iterative Agent-based simulations in a way that the analytical FRF of the occupied structure matches the corresponding experimental FRF measured from a series of walking tests on a lightweight floor structure. It was found that a 2DOF human model with modal frequencies 4.3Hz and 11.0Hz and modal damping ratios of 19% and 0.52% for modes one and two, respectively, generates the least error in estimated FRFs. The validation of the identified 2DOF model on a post-tensioned concrete footbridge with different walking traffic regime showed that the model can estimate the natural frequency and damping ratio of both vibration modes of the occupied structure with maximum error of 0.4% and 6%, respectively. This is the first study of its kind that proposes a physical model of a walking human that consistently simulates the dynamics of a walking human for different structures and different modes of a structure, and addresses the discrepancy in the identified parameters of the SDOF walking human model in the literature.

Evaluation of Seismic Deflection Amplification Factor for Buildings Utilizing Cold-Formed Steel–Framed Shear Walls

The seismic deflection amplification factor for a series of archetype buildings utilizing cold-formed steel–framed shear walls was investigated. A total of 118 archetype buildings were considered, of which 83 employed cold-formed steel–framed shear walls with steel sheet sheathing, and 35 employed wood structural panel sheathing. Using nonlinear equivalent single-degree-of-freedom (SDOF) models, nonlinear time-history analyses for the 118 archetype buildings subjected to the 44 FEMA P-695 earthquakes were conducted. In these SDOF models, the actual response of shear walls obtained from experimental studies was used to define the nonlinear material parameters. The AISI S400 deflection expression was evaluated using experimental data, and numerically predicted deflection amplification factors were calculated and compared against that recommended in current practice. The results show that current practice may overestimate expected deflections. To address this issue, and to simplify design, a linearization of the AISI S400 deflection expression is recommended. With appropriate choice of the force level, the linearized AISI S400 deflection expression leads to acceptable drift predictions with the currently employed deflection amplification factor.

A single degree of freedom model for cracked beam

This paper presents a simplified model of cracked beam by single-degree-of-freedom system. Equivalence between the beam and SDOF models means that they have the same fundamental natural frequency and similar frequency response functions (FRFs). Similarity of FRFs is checked by using the frequency-domain assurance criterion acknowledged herein as spectral similarity index (SSI). Finally, FRFs of both the cracked beam and its simplified SDOF model have been examined versus crack location and depth using the so-called spectral damage index (SDI). Numerical results show that SDI is significantly sensitive to crack and could be used as a novel indicator for crack detection in beam by measurements of frequency response functions.

Experimental investigation into the transverse impact performance of high-strength circular CFST members

Both high-strength steel and high-strength concrete are gaining increasing use in the construction industry. At the same time, the benefits of composite construction are also being increasingly recognized and exploited. In the present study, high-strength steel and high-strength concrete are considered in combination in high-strength concrete-filled high-strength steel tubular (HSCFST) members, with a focus on their behaviour under transverse impact loading, resistance to which is important for resilient infrastructure. Tests on thirteen HSCFST specimens and six reference CFST specimens under drop weight impact loading are presented. Hot-finished high-strength S890 steel sections were employed for the outer tubes of the HSCFST specimens, while hot-finished S355 steel tubes were used for the reference CFST specimens. Two core concrete strengths of approximately 60 MPa and 100 MPa were considered. The instantaneous impact force and deformation histories of the specimens were recorded at high frequencies throughout the impact process. A single-degree-of-freedom (SDOF) model was developed and used to facilitate the analysis of the experimental results. The dynamic moment capacities of the test specimens were obtained using the developed SDOF model and a dynamic increase factor (Rd) was introduced to quantify the enhancement in moment capacity relative to the static values. The influence of the test parameters on the dynamic increase factors was then analysed. The test results indicated that Rd is positively correlated with the impact velocity and negatively correlated with the steel grade, steel ratio and specimen length, while being insensitive to the concrete strength.

Source contribution analysis of vehicle pass-by noise using a moving multi-band model based OPAX method

The conventional operational path analysis with eXogeneous inputs (OPAX) method has been widely used to investigate the vibration transfer paths. However, the single degree of freedom model is not available for the load identification of acoustic sources due to their unknown source behavior. And the alternative multi-band (MB) model can only offer a deterministic estimation to the load parameters. To assess the acoustic source contribution of vehicle pass-by noise (PBN), this paper develops the OPAX method by introducing a moving multi-band model (MMB). In this method, the vehicle acoustic sources are simplified as a combination of a series of point sources. The source loads are firstly parametrically modelled by MB models. Then the load parameters are solved several times in the MMB and estimated by the index of coefficient of variation, which suppresses the effect of nonlinear random noise and improves the load identification accuracy of acoustic sources. Finally, the source contributions to PBN at receivers can be derived. Both numerical studies and experimental validations are implemented to examine the performance of the proposed method. Results show that compared to the conventional OPAX method, the proposed method can provide a higher accuracy for the source contribution analysis of PBN.

Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system

Nonlinear Single degree of freedom (SDOF) systems are crucial for understanding the behavior of various real-world systems, designing and analyzing mechanical and dynamical systems. In this article we have modified the SDOF model by introducing the nonlinearity as well as damping with/without external force and applied the homotopy perturbation method (HPM) to explain its various phenomena. We have compared the analytical solutions obtained via the HPM to the provided numerical results by the fourth-order Runge-Kutta (RK4) method for verifying the precision and validity of the solutions. The presentation and explanation of the dynamic results can add a new dimension to the research field by influencing the importance of nonlinear SDOF systems.

Applied element modelling of unreinforced and reinforced concrete masonry walls under blast loading

Masonry walls are the first defense line against exterior explosions in buildings; however, no adequate guidelines are currently available for the analysis and design of these walls under such extreme loading. In this paper, a comprehensive applied element model (AEM) is developed and validated to investigate the behaviour of both reinforced (RMWs) and unreinforced (URMWs) concrete masonry walls under blast loading. A single degree of freedom (SDOF) model is also developed to estimate their pressure-impulse diagram for concrete masonry walls. The developed models are utilized to conduct a parametric study by analyzing 30 walls with different design parameters (boundary conditions, dimensions, compressive strength, as well as reinforcement ratio) under various blast loadings. Crack pattern, fundamental period, support rotation, as well as pressure–displacement and pressure-impulse diagrams for the analyzed walls were compared. The boundary conditions and compressive strength had the most and least significant influence on the behaviour of walls. The two-sides-supported walls experienced severe cracks compared to the four-sides-supported walls. Fundamental period decreased by 74% when the wall thickness increased from 100 mm to 250 mm for either URMWs or RMWs. Resistance of a wall with dimensions of 3 × 7 m was less than that of a 5 × 2 m wall by 81% and 77% for URMW and RMW, respectively. The SDOF model showed that URMWs and RMWs with dimensions of 5 × 2 m had the maximum pressure-impulse response limit compared to the other walls. The average increase in the support rotation was 200% and 380% when the charge weight increased from 5 kg to 15 kg for URMWs and from 15 kg to 150 kg for RMWs, respectively. Using reinforcement ratio greater than 0.85% considerably improved the performance of RMWs.

Along-wind and across-wind coupling analysis of high-rise buildings: Modeling and parameter identifications

High-rise buildings, with similar natural frequencies of sway modes in two orthogonal directions, can have coupled vibrations when exposed to strong wind exhibiting complex fluid-structure interaction and aero-elastic effects. This paper studies this coupled vibration with proposing of a mathematical model for both the in-plane and out-of-plane self-excitation force. A single-degree-of-freedom (SDOF) and two-degree-of-freedom (2DOF) aero-elastic square section high-rise building models are tested in wind tunnel to provide data for the analytical modeling. A new strategy to estimate the aerodynamic parameters and self-excitation forces is proposed. Parametric studies show that the self-excitation forces of the SDOF and 2DOFs models have significant difference. The out-of-plane self-excitation force acting in the along-wind direction induced by the across-wind vibration can become dominating inducing large amplitude vibration in the along-wind direction. The wind induced coupling vibrations for other tall buildings can similarly be studied in detail with the proposed strategy.

Can a nonlinear quasi-zero-stiffness spring improve the ride quality of a vehicle?

The paper examines the possibility of using a nonlinear Quasi-Zero-Stiffness (QZS) spring in a vehicle suspension. The response of a Single Degree of Freedom (SDOF) model to harmonic base excitations is obtained which may be unstable and unbounded depending on the excitation level and the damping ratio. This is followed by obtaining the response of the SDOF model to harmonic base excitation with an amplitude that is varying according to a road profile spectrum. Such dependency of the excitation amplitude to the frequency changes the qualitative behaviour of the nonlinear system and the responses would be always bounded. The QZS spring also improves the isolation of the system. Transient responses of a quarter car model with a QZS suspension to a road hump and random road profile are investigated. The maximum acceleration of the vehicle with the QZS suspension passing over the speed hump is considerably lower than a vehicle with a conventional linear suspension. Wb weighted RMS of acceleration (BS 6841-1987) is also lower by as much as 14% for a vehicle with the QZS suspension travelling at 30 km/h on a class E road compared to its linear counterpart.

Vibration Performance Analysis and Multi-Objective Optimization Design of a Tractor Scissor Seat Suspension System

The combination of characteristic parameters is the key and difficult point to improving the vibration attenuation of scissor seat suspension. This paper proposes a multi-objective optimization method based on entropy weight gray correlation to optimize the combination of characteristic parameters with better vibration attenuation. The differential equation of seat suspension motion is derived through mechanical analysis, and a simplified driver seat suspension single degree of freedom model is constructed. The range of spring stiffness and damper damping is calculated theoretically. Through main effect analysis and analysis of contribution, the main influencing factors of seat suspension vibration attenuation are studied, and the influence correlation of the main factors is analyzed. On this basis, the spring stiffness and damper damping are taken as control variables, and the upper plane acceleration, displacement, and transfer rate of the seat suspension are taken as optimization objectives. The Optimal Latin Hypercube Sampling (OLHS) was used to sample the Design of Experiments (DoE), fit the RBF surrogate model, and screen the optimal solution based on the MNSGA-II algorithm and entropy weight gray relation ranking method. The comparative analysis of the performance before and after optimization shows that the vibration reduction performance response indexes of the acceleration, displacement, and transmissibility of the optimized seats are increased by 66.41%, 2.31%, and 8.19%, respectively. The design and optimization method proposed in this study has a significant effect on the vibration reduction of seats, which provides a reference for the optimization of the vibration reduction performance of seat suspension.

Damage boundaries on shock response spectrum based on an elastic single-degree-of-freedom structural model

Damage boundary based on a single-degree-of-freedom (SDOF) structural model under shock environment is studied in the present paper. Shock environment applied to the SDOF structure is characterized by different shock-waveforms. For an SDOF model with small damping, its shock damage boundary consists of three characteristic lines intersecting at a point, i.e. the isodamage absolute acceleration asymptote, pseudo velocity horizontal line and relative displacement asymptote in different frequency regimes. The upper and lower damage bounds on 4-coordinate pseudo velocity shock response spectrum (PVSRS) graph are proposed, which are associated with generic shock-waveforms with sufficiently large and small shock-waveform coefficients, respectively. The validity of the upper and lower damage bounds proposed in this study is demonstrated using an idealized mass-plate model with pins subjected to shock loads characterized by various shock-waveforms. Particularly, the lower damage bound underpins the use of PVSRS ‘damage metric’ as a rule-of-thumb in various engineering fields. It also shows that the proposed method is suitable for real shock environment.

Influence of Inertia on the Dynamic Compressive Strength of Concrete.

The rate sensitivity of concrete material is closely related to the inertia and viscous effects. However, the effect of inertia on the dynamic strength of concrete remains unclear. In this paper, digital image correlation technology was applied to study the strain variation of dry and saturated concrete with different loading rates. The test results indicated that the strain gradually decreased with the distance from the load end, and the strain gradient around the load region increased with the strain rate, especially for saturated concrete. Then, a single degree of freedom model was established to evaluate the dynamic compressive strength of elastic concrete. The calculated results indicated that the influence of inertia on the dynamic increase factor (DIF) was negligible for concrete within a low strain rate. When the strain rate is larger than 100/s, the inertial effect on the strength of concrete should be considered. After that, a quasi-static concrete damaged plasticity (CDP) model was employed to simulate the influence of inertia on the stress distribution and axial reaction force at the loaded end of concrete under different rates of compressive loading and verified with experimental results. The results obtained in this study indicated that the dynamic nominal strength of concrete obtained from the tests could not be directly used for structural analysis which may overestimate the effect of inertia on the dynamic response of the structure.

Vibro-Impact Response Analysis of Collision with Clearance: A Tutorial

A collision with clearance causes obvious nonlinearity in structures, and dynamic response analysis plays an important role in predicting the mechanical performance of the structure. The general form of the nonlinear dynamic equation of a structure and the clearance modeling method are introduced, and the clearance-caused nonlinear term is expressed by nonlinear impact forces. Different clearance collision models of local nonlinear structures are presented. The relationships between different impact forces and clearances are analyzed by two rigid sphere models. The solution methods of the nonlinear dynamic equation are compared by a vibro-impact response, such as the Newmark-β method combined with the Newton–Raphson method, generalized α method and precise integration method. The single degree of freedom model is adopted to compare the efficiency of the different numerical integration algorithms. Taking the beam structure model as a case study, the accurate nonlinear collision model with clearance is established by using the impact force model with high accuracy, and the accuracy of the model is verified by comparing the reference model with the numerical model.

Development of pressure-impulse diagram to predict the damage of simply supported RC beams under close-in explosion

This paper develops the pressure-impulse (P-I) diagram to conduct damage assessment of simply supported reinforced concrete (RC) beams under close-in explosion based on the equivalent single-degree-of-freedom (SDOF) model. Firstly, eleven close-in explosion tests are carried out with different charge weight, scaled distances and offset distances of explosives relative to the mid-span of RC beams. Test results show that the maximum rotation angle can characterize the comprehensive spalling/bending damage under asymmetric load, therefore it is selected as the damage criteria. Then, the equivalent SDOF model is improved by considering the influence of asymmetric load on flexural resistance, and is verified through the test results. Both test results and SDOF model show that increasing the offset distance results in asymmetric bending failure and reduces peak deflection. Through comparative analysis, the equivalent uniform load is chosen to be the axis of P-I diagram. The asymptotes and shape parameters of P-I curves are related to the offset distance and load uniformity. The relationship is derived through curve fitting and validated with the test results and the calculated results of SDOF model. This work expands the application scope of P-I diagram, which contributes to a more comprehensive damage assessment of RC structures under close-in explosion.

Seismic fragility analysis of concrete pile-supported wharves based on single-degree-of-freedom model

This study proposes a new procedure to predict the seismic fragility curves of concrete pile-supported wharf based on an equivalent single-degree-of-freedom (SDOF) model. The SDOF model is utilized to substitute the practical wharf structure during probabilistic seismic demand analysis for reducing the computational cost and complexity. The trilinear backbone cure and Pivot hysteresis model, whose parameters are determined by the pushover analysis of actual wharf, are incorporated in the SDOF model to represent hysteresis characteristics and strength degradation of wharf. To validate the reasonability of the proposed procedure, two case studies are conducted by comparing the difference of fragility curves developed from actual wharf and its equivalent SDOF model based on the cloud method. The results show that the curves are located very closely on the graph, and the curve of SDOF model is generally above that of actual wharf, which indicates that the SDOF model may provide a conservative assessment for fragility analysis of wharf.

Structural control of high-rise buildings subjected to multi-hazard excitations using inerter-based vibration absorbers

High-rise buildings located in strong wind and earthquake-prone regions are exposed to multiple hazards that must be properly considered to prevent considerable socio-economic losses. In the present study, the Tuned Mass Damper-Inerter (TMDI) is adopted for the structural control of high-rise buildings under wind and earthquake excitations by exploiting its inherent multiple-mode control effects. A generalized multi-degree-of-freedom (MDOF) model of TMDI-controlled structure is established first and compared with the commonly-adopted generalized single-degree-of-freedom (SDOF) model. Analytical investigation on the multiple-mode control effects of TMDI demonstrates that the damping effects weighted by mode coordinate differences at two terminals of TMDI are mainly responsible for its multiple-mode control effects. Based on a real high-rise building project, analysis errors induced by the generalized SDOF model against the more accurate generalized MDOF model are quantified by a comprehensive Monte Carlo simulation, which suggests the necessity of adopting numerical search for parametric design of TMDI under most circumstances. Therefore, the parameters of TMDI and its two competitors, i.e., Multi-Tuned Mass Damper (MTMD) and Multi-Tuned Mass Damper-Inerter (MTMDI), are optimized using a genetic algorithm, assuming both wind- and earthquake-induced responses as performance objectives. Time and frequency domain analyses demonstrate the efficiency of utilizing the multiple-mode control effects of a single TMDI for multi-hazard design. The trend of the equivalent damping ratio of TMDI-controlled structure on an arbitrary mode is analyzed to identify the design parameters affecting most the TMDI multiple-mode control effect at last.