Multi-degree-of-freedom Systems Research Articles

Open Ground Story Mid-Rise Buildings Represented by Simplified Systems

An improved methodology for the condensation of Multi-Degree-Of-Freedom (MDOF) systems to equivalent Two-Degree-Of-Freedom (2EDOF) systems is presented. The methodology is applied to mid-rise buildings with Open Ground-Story and verified by means of Nonlinear Time History Analyses. The buildings studied are divided into two main segments: (a) ground story, whose stiffness and lateral strength are both provided only by reinforced concrete moment-resisting frames; and (b) from the second story to the roof, whose stiffness and lateral strength are provided by confined masonry walls. The proposed methodology allows us to do the following: (a) get the closest approximation to the real behavior of the MDOF system through the use of hysteresis rules with strength and stiffness degradation in the simplified system; (b) analyze the behavior of an OGS whose lateral stiffness is lower than the stiffness of the stories above; and (c) identify in which of the two subsystems (either the ground story with reinforced concrete frames or the second story with masonry) the maximum seismic demand of non-linear behavior occurs. For most of the cases studied and different scenarios of non-linear behavior, the 2EDOF simplified system reasonably approximates the MDOF system’s response; however, when a local failure in an upper story causes the collapse mechanism, the 2EDOF system does not adequately approximate the response of the MDOF system.

Relationships between ground motion parameters and energy demands for regular low-rise RC frame buildings

AbstractThe present study aims to analyze the correlation between ground motion parameters and energy demands of low-rise RC buildings without shear walls. Two regular 4- and 7-story residential buildings were seismically designed to represent low-rise RC buildings. In order to establish the requirements of single degree of freedom (SDOF) systems as well as multi degree of freedom (MDOF) systems, the dynamic features of “equivalent” SDOF systems were defined by using MDOF systems. The correlation of 20 ground motion parameters (GMPs) of 44 records with the energy demands obtained from a total of 176 nonlinear time history analyses was examined for the SDOF and MDOF systems within the scope of this study. The ground motion parameters (GMPs) were taken as intensity measures (IMs). In contrast, maximum input energy was used as a demand measure (DM) and these energy values were normalized with the masses of the buildings for cases, where the buildings were evaluated together. Parameters related to acceleration and velocity are generally found to yield better results in comparison to the ones related to frequency and displacement. Velocity Spectrum Intensity (VSI) and Arias Intensity (Ia) were obtained to have the highest correlation values as a single parameter. This study suggested new equations by combining multiple ground motion parameters for SDOF and MDOF systems to better reflect damage potential than a single parameter. The use of multiple parameters in combination results in better correlation coefficients.

In-plane free vibration analysis of multi-folded beam structures

Folded beam structures have been widely used in mechanical engineering, aerospace engineering, and other fields due to their flexibility and extensibility in applications. Although the components of a folded beam are simple beam structures, building an accurate dynamic model is complex due to its multi-degree-of-freedom, multi-mode, and close-mode features. This study presents a novel method to analyze in-plane free vibration of a multi-folded beam structure with arbitrary numbers of beam segments with different beam lengths and folding angles. The governing equations of the system can be easily obtained based on the beam theory and the work-energy principle. However, the key issue is how to solve the eigenvalue problem of a multi degree of freedom system defined by a 6 n × 6 n matrix. As the coefficient solution vector of the modal function is unknown, solving equations defined by high-order determinants involving variables has considerable complexity. In this work, the intricate matrix is initially turned into a single matrix of block Hessenberg form using numerous "6 × 6" block square matrices. Furthermore, by combining the characteristics of the block Hessenberg matrix with the approach to solve block determinants, a considerable reduction in the determinant's size can be attained, diminishing it from a dimension of 6 n × 6 n to 6 × 6. This procedure markedly simplifies the computational intricacies involved in the analysis. Finally, four kinds of multi-folded beams, namely L-shaped beams, Z-shaped beams, C-shaped beams, and four-segment folded beams with arbitrary lengths, folding angles, and numbers of beam segments are considered and studied, and the mode shapes and the influence of geometric parameters on the first three natural frequencies are discussed in detail. The results are also verified by the finite element method (FEM), and good agreement is found between the two methods. The results show that the proposed method can offer a unified framework for the in-plane vibration analysis of multi-folded beam structures.

A new probabilistic long-period critical excitation procedure for isolated building structures

The present paper develops a new procedure for finding the most unfavorable long-period excitation for isolated building structures based on the probabilistic critical excitation method. The upper bound of earthquake input energy per unit mass is considered the problem’s constraint, and a modified power spectral density (PSD) function is proposed to conform to the new constraint. It is shown that the new constraint can be used as a criterion to estimate the possible range of input energy of similar earthquakes. In addition, the new PSD function can control the excitation’s intensity. Finally, three base-isolated shear buildings modeled as multi degree of freedom systems with nonproportional damping are considered, and the critical excitations for each model are found based on the proposed method. To examine the reliability of the results, three actual accelerograms with a considerably high level of total input energy are selected as benchmarks and linear dynamic analyses are conducted using these accelerograms along with the generated critical excitations. The results show that the generated long-period excitations can better estimate the structural behavior (i.e., maximum displacement, drift, and absolute acceleration of stories) than the actual benchmark accelerograms.

An efficient framework for structural seismic collapse capacity assessment based on an equivalent SDOF system

Assessing the structural collapse capacity efficiently and accurately is a key issue in earthquake engineering. Here, an efficient framework for structural seismic collapse capacity assessment based on an accurate equivalent single-degree-of-freedom (ESDOF) system is proposed. The corresponding ESDOF system is calibrated by matching the cyclic static pushover (SPO) curves of a more complex multi-degree-of-freedom (MDOF) system (e.g., high-fidelity finite element models) through a meta-heuristic optimization method. In this way, both the backbone curve and hysteretic characteristics (the stiffness and strength deterioration, and pinching behavior) of the complex MDOF system are considered in the corresponding ESDOF system. Then, incremental dynamic analysis (IDA) is carried out to assess the structural seismic collapse capacity using the ESDOF system instead of the corresponding MDOF system to improve the computational efficiency. The efficiency and accuracy of the proposed framework have been validated through three case studies, including a bare reinforced concrete (RC) frame, a steel frame, and an infilled wall RC frame. These results affirm that the proposed framework can accurately and efficiently assess the collapse capacity of low-rise bare and infilled RC frame structures, as well as steel frame structures.

MDOF stochastic stability analysis and applications to a coupled rotating shaft system

Stochastic stability analysis and generalization of moment Lyapunov exponents of multi-degree-of-freedom (MDOF) systems under white noise excitation were carried out using the transformation method and the perturbation method. Such systems represent coupled dynamic models exposed to stochastic disturbances and they are very common in engineering practice. The analysis of individual parts of a mechanical system does not give a clear picture of its real behavior as a whole, so the consideration of dynamic couplings is of great importance. This study proposes a generalized set of new transformations for MDOF systems that are necessary for obtaining Itô’s equations needed for further consideration of stochastic stability. The eigenvalue problem for moment Lyapunov exponents is derived and then solved by the method of perturbation. The generalized procedure is applied in an example of a rotating coupled shaft system whose dynamic behavior should be analyzed with at least four generalized coordinates. Similar models have so far been analyzed with limitations on SDOF and 2DOF systems that do not fully correspond to realistic models. The methodology demonstrated in the study could be applied more broadly, as it has the potential to analyze the dynamic stability of even more intricate or diverse systems whose dynamics have not been previously studied.

Structural pounding of three adjacent multi degree of freedom system under subsurface blast and seismic action by considering and ignoring the effect of soil structure interaction

A lack of spacing between the adjacent structures is mostly observed in metropolitan cities due to more land cost, which has led to more serious problem called as structural pounding in many recent earthquakes. Large impact forces and short acceleration pulses are mostly noticed during the structural pounding, and which is hazardous to structural health of the buildings involved in this phenomena. In the present paper three adjacent multi-degree-of-freedom linear elastic systems subjected to seismic and subsurface blast actions are analyzed by considering and ignoring the effect of soil structure interactions. Three dimensional buildings are analyzed and designed for deciding the column and beam sizes in SAP 2000 NL software considering all possible load combinations of dead load, live load and earthquake load as per IS 1893:2016 Part I. The subsurface indirect method is used for studying the effect of soil structure interaction. The SSI spring stiffness’s and damping constants are calculated as per FEMA 356. These buildings are converted into the lumped mass model as per the principles of dynamics of structures. The entire analysis of three adjacent lumped mass models with insufficient separation is carried out in time domain form using MATLAB programming. The outputs are measured in the forms of top storey displacement time history plots, top floor level pounding force time history plots and base shear force time history plots. From the outputs, it is noticed that the SSI amplifies the displacement response of building than that of fixed base system. SSI also overrates the pounding response of structure. Blast force response is much smoothens than seismic response of structure.

A polynomial extrapolation-based wavelet-Galerkin method for dynamic response reconstruction

In this work we present a single scale wavelet-Galerkin method (WGM) for obtaining the linear dynamic response of single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems, the equation of motion is discretized using a polynomial extrapolation-based time stepping scheme. The aim of the study is to investigate the accuracy of the dynamic response resulting from the application of the proposed method and to explore the Daubechies wavelet types that can be used for unconditional stability. In our work we solve only for the scaling function coefficients of the displacement response, the scaling function coefficients of the velocity and the acceleration responses are obtained from the solved scaling function coefficients of the displacement function and the pertaining connection coefficients. The classical central differences method (CCDM) is used to generate consistent values of the velocity and the acceleration response to replace the distorted ones at the edges of the time interval. Stability analysis shows that the method using D6 wavelets is unconditionally stable for the practical range of the product of the time step and the frequency of vibration. An application of a SDOF system with initial conditions shows that for D6 wavelets the calculated responses at an adequate approximation level correspond well with those obtained using Newmark’s average acceleration method (AAM). The method is also applied to a MDOF system without performing a modal analysis, the resulting dynamic response is found to be accurate enough to be compared to that of the mode superposition method used by the STAAD pro program.

Simplified Estimation Method of Plastic Energy Dissipation for MDOF Systems Using Force Analogy Method

Plastic energy dissipation is a key factor in the response of inelastic structures subjected to seismic input, and is often regarded as a primary source of structural damage due to the inelastic deformation of structural components. Accurately predicting a structure’s plastic energy dissipation is essential for efficient energy-based design and seismic assessment. While a single-degree-of-freedom (SDOF) system provides a simple and effective method for estimating plastic energy dissipation, few studies have explored the use of nonlinear analysis methods or equivalent SDOF systems for this purpose. Based on the principle-of-force-analogy method, firstly, the formulas of plastic energy dissipation of a multi-degree-of-freedom (MDOF) system and its equivalent SDOF system were established. Secondly, two estimation methods of plastic energy dissipation of MDOF systems were proposed. Finally, numerical simulations were performed on several multi-story and high-rise structures with varying heights and spans to compare the plastic energy dissipation of MDOF systems and the equivalent SDOF systems of different modes. The simulation results demonstrate that the proposed methods and formulas accurately estimate plastic energy dissipation in multi-story and high-rise structures, while also requiring fewer calculations and less storage.

Seismic response of slender MDOF structures with self‐centering base shear and moment mechanisms

AbstractAs the amplification of seismic force demands due to higher‐mode effects on tall buildings is increasingly recognized as a design challenge, a number of high‐performance systems have been proposed to limit such effects through combined base shear‐limiting and moment‐limiting dual mechanisms. To better understand the influence of these dual base systems on the overall seismic behavior of tall buildings, this paper presents a study on the seismic response of Multi‐Degree‐of‐Freedom (MDOF) systems incorporating combined base shear‐limiting and moment‐limiting mechanisms. For an MDOF system with a given initial period and base strength level, the base shear‐limiting mechanism is defined by a shear strength factor, an inelastic stiffness parameter, and an energy‐dissipation parameter, while the base moment‐limiting mechanism is defined by a moment strength factor. To determine the influence of these parameters on the overall seismic responses of MDOF structures, a comprehensive parametric study was conducted, and the results were presented and discussed in terms of base displacement and rotation demands, seismic force demand amplification at the base and along the height, peak floor acceleration, peak roof drift, and absorbed energy. The numerical modeling methodology used in the parametric study was validated against 200 small‐scale shaking table tests of a scaled MDOF specimen with a base shear and moment dual‐mechanism system and was used to model MDOF structures that are representative of tall buildings with initial periods ranging from 1.0 to 10 s and having various base‐mechanism properties. The parametric study was then conducted using an ensemble of 20 Ground Motions (GM) scaled to three code‐specified seismic hazard levels for Los Angeles, California. The results of this study can be used to facilitate the design of base shear and moment dual‐mechanism systems for mitigating higher‐mode effects and enhancing the seismic resilience of tall buildings.

Residual storey drift estimation of the MDOF system with the weak storey under seismic excitations using the BP network

This paper focuses on estimating residual storey drifts of multi-degree-of-freedom (MDOF) systems with the weak storey due to seismic excitations. According to the distribution of yield strength coefficients, MDOF systems are categorized into two types: a uniform MDOF system and a non-uniform MDOF system. Through elasto-plastic time history analysis, typical MDOF systems were analyzed considering the P-Δ effect under 60 earthquake records free of near-fault effects. The influences of various structural parameters on distribution characteristics of residual storey drifts along the height of the MDOF systems were investigated in detail. Finally, a back propagation (BP) network model was proposed to estimate residual storey drifts of the MDOF systems with the weak storey under seismic excitations. The results indicate that the maximum residual storey drift usually occurs at the weak storey for a non-uniform MDOF system while at the bottom for a uniform MDOF system. In addition, the larger the overall yield strength coefficient, the yield strength modification factor or the post-yield stiffness coefficient, the smaller the residual storey drift of the MDOF system on the whole. Moreover, the proposed BP model can sufficiently consider the parameters affecting residual storey drifts of the MDOF system with the weak storey, while having a good accuracy and a high efficiency in estimating the residual storey drifts.

Stochastic dynamic analysis of nonlinear MDOF systems with chaotic motion under combined additive and multiplicative excitation

Stochastic response and global dynamic analyses of structures with chaotic motion are challenging issues, especially for nonlinear multi-degree-of-freedom (MDOF) system excited by combined additive and multiplicative excitation. In this paper, a novel direct probability integral method (DPIM) is extended to address these challenges. Firstly, the probability density integral equation (PDIE) of MDOF system under combined excitation is established. By using the DPIM to solve the decoupled deterministic dynamic equation and PDIE successively, the stochastic responses are then achieved, and the stochastic bifurcation of system under combined excitation is explored. Moreover, the important equivalent relationship between PDIE and Dostupov–Pugachev differential equation is derived, exhibiting the superiority of PDIE for MDOF system under additive and multiplicative excitation. Due to the lack of effective numerical tool for global dynamic analysis of nonlinear MDOF stochastic system, another aim of this study is to propose a DPIM-based strategy to attack this problem from the view of probability. As the generalized stochastic basin, the ϵ-committor is introduced, and global integrity measure (GIM) is utilized for evaluating the stability of stochastic basin quantitatively. Finally, two examples of nonlinear systems under combined excitation, including nonlinear MDOF coupled ship model with chaotic motion under random oblique wave, demonstrate the effectiveness of DPIM. It is shown that the safety basin of system under combined excitation can be effectively described in a probabilistic way. The dramatic effect of initial disturbance on stochastic ship system is revealed, i.e., the stochastic safety basin is broken up to a series of discretized regions with increasing of intensity of initial disturbance, resulting in the decreasing of system stability.

Stochastic seismic analysis of structures with nonlinear eddy current dampers

Eddy current damper (ECD), a contactless energy-dissipating device, is applying to control the vibration induced by earthquake and strong wind in civil structures. Combining with motion magnification mechanisms improves the damping effect of the ECD while significantly strengthens its nonlinearity. The response of single degree of freedom and multi-degree of freedom system with ECDs under a stationary stochastic earthquake characterized by the power spectral density function is evaluated using the stochastic linearization technique and expressions of the equivalent linear damping coefficient based on force criterion and energy criterion have been found, respectively. Comparisons with results obtained by Monte Carlo simulations confirm that for the nonlinearity of eddy current dampers the force-based criterion stochastic linearization technique gives accurate estimation.

Closed-form critical response of elastic-plastic shear building with viscous damper under pseudo-double impulse for simulating resonant response under near-fault fling-step motion

A set of approximate closed-form solutions of the maximum interstory drifts under the critical pseudo-double impulse (PDI) is derived for non-proportionally damped multi-story shear building models with bilinear hysteresis. The use of PDI and the closed-form solutions efficiently and accurately enable the capture of the critical responses of elastic-plastic multi-degree-of-freedom (MDOF) systems under the one-cycle sine wave, which substitutes for the main part of near-fault fling-step motions. The formulation of the closed-form maximum interstory drifts is based on the energy balance law. In the formulation, a quadratic function approximation of the damping force-interstory drift relation is introduced together with an updated mode-controlled energy-based approach (UMEA). While UMEA was proposed in the previous paper to derive the approximate maximum interstory drifts of undamped elastic-plastic MDOF systems under the critical PDI, it is extended so that it can be applied to non-proportionally damped MDOF systems. It is demonstrated that the proposed method can estimate the maximum interstory drifts of elastic-plastic non-proportionally damped MDOF systems under the critical PDI and the corresponding one-cycle sine wave with high accuracy. The estimation by the proposed method can be conducted much more efficiently and stably than the time-history response analysis (THRA). It is also shown that the hysteretic energy dissipation is not large enough to reduce the maximum interstory drifts. Finally, it is demonstrated that the proposed method can accurately estimate the maximum interstory drifts of elastic-plastic moment-resisting frames with viscous dampers under the resonant one-cycle sine wave.

Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

TL-φ algorithms are newly developed explicit structure-dependent integration algorithms utilized for solving the temporally discretized equations of motion. In contrast to the existing algorithms, the most significant improvement of TL-φ algorithms is in diminishing the amount of period errors by introducing a precorrection coefficient φ into the integration parameters of TL-φ algorithms, which is related to the critical frequency of a system. In the previous work, the fundamental frequency of the system is deemed to be the critical frequency, so that φ is a constant scaling corresponding to the fundamental frequency for both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. However, for a MDOF system, the first mode may not contribute to the total response more than other ones under a given external excitation, calculating φ only by the fundamental frequency will underestimate the contribution of the higher-frequency modes to structural dynamics. In this paper, choices of the critical frequency for φ when applying TL-φ algorithms to MDOF systems are investigated thoroughly. By considering the initial structural properties of the system and the frequency characteristics of the external excitation simultaneously, a calculation criterion of φ for MDOF systems under specific external excitations is proposed. Four numerical examples with different initial structure properties and loading conditions are designed, and the results demonstrate that the proposed criterion can be potentially used to solve structural dynamic problems of MDOF systems with a more desirable numerical dispersion performance.

A quadratic Wiener path integral approximation for stochastic response determination of multi-degree-of-freedom nonlinear systems

A Wiener path integral (WPI) technique is developed for determining the stochastic response of multi-degree-of-freedom (MDOF) nonlinear systems. Specifically, the nonlinear system response joint transition probability density function (PDF) is expressed as a WPI over the space of paths satisfying the initial and final conditions in time. Next, a functional series expansion is considered for the WPI and a quadratic approximation is employed. Further, relying on a variational principle yields a functional optimization problem to be solved for the most probable path, which is used for determining approximately the joint response transition PDF. It is shown that compared to the standard (semiclassical) WPI solution approach, which accounts only for the most probable path, the quadratic approximation developed herein exhibits enhanced accuracy. This is due to the fact that fluctuations around the most probable path are also accounted for by considering a localized state-dependent factor in the calculation of the WPI. Furthermore, the PDF normalization step of the most probable path approach is bypassed, and thus, probabilities of rare events (e.g., failures) can be determined in a direct manner without the need for obtaining the complete joint response PDF first. The herein developed technique can be construed as an extension of earlier efforts in the literature to account for MDOF systems. Several numerical examples are considered for demonstrating the accuracy of the technique. These pertain to various dynamical systems exhibiting diverse nonlinear behaviors. Comparisons with pertinent Monte Carlo simulation data are included as well.

Numerical Experiments on the Optimum Design and Location of a Passive Vibration Absorber Applied to Multi Degrees of Freedom System Under Base Excitation

Numerical experiments are used to investigate the optimal design parameters and the location of a passive vibration absorber and its application to reduce resonant vibration amplitudes of a base excited, discrete, Multi Degrees of Freedom (MDOF) vertical system. Attaching a passive vibration absorber (simple undamped resonator) to the system’s end mass (top mass) is preferable when dealing with the vibrations of multi degrees freedom systems. As the absorber is moved closer to the primary system's farthest end, the amplitude reduction results get better. The numerical analysis is less sensitive to slight variations in mass ratio than minor variations in the absorber’s natural frequency, according to perturbation analysis on a two degree of freedom system. Introducing damping to the used passive vibration absorber has no discernible effect on the absorber’s effectiveness.

Stochastic dynamic analysis of nonlinear MDOF systems under combined Gaussian and Poisson noise excitation based on DPIM

Random vibration analysis of structures subjected to combined Gaussian and Poisson white noise excitation is a challenging issue. In this paper, a novel direct probability integral method (DPIM) is suggested to address stochastic responses and dynamic reliability of nonlinear multi-degree-of-freedom (MDOF) systems under combined excitation. Firstly, probability density integral equation (PDIE) of MDOF system under combined excitation is derived based on the principle of probability conservation. Then, DPIM is proposed to achieve the probability density function of stochastic response by solving deterministic motion equation of MDOF system and PDIE in sequence. To solve PDIE, two techniques, i.e., partition of input probability space and smoothing of Dirac delta function, are introduced. From the perspective of probability conservation, furthermore, the equivalent relationship between the PDIE and the corresponding probability density differential equation of a Markov system under combined Gaussian and Poisson noise is established. Since Dirac delta function is analytically integrated as Heaviside function, the first-passage dynamic reliability of MDOF system under combined excitation is readily evaluated by introducing the extreme value mapping of stochastic response. Finally, two nonlinear MDOF systems, including multiple-span bridge under combined vehicle load and non-stationary seismic excitation, are solved. Results demonstrate that the proposed DPIM is effective for random vibration and dynamic reliability analyses of MDOF structures excited by combined Gaussian and Poisson noise, and considering the randomness of vehicle load and seismic excitation simultaneously in the bridge design benefits the safe operation of bridge.

Generalization of Duhamel's integral to multi-degree-of-freedom systems

This article presents a generalization to multi-degree-of-freedom (MDOF) systems of the classical convolution formula for single-degree-of-freedom (SDOF) systems that is widely known as Duhamel's integral. Remarkably, the extended convolution formula for a MDOF system is ultimately cast directly in terms of the system stiffness, mass and damping matrices in a manner that fully parallels the SDOF case, and without resorting to real or complex modal superposition. Still, although the proof based on the use of the Laplace transform might suggest that the obtained formulae are applicable to any type of viscous damping matrix, a more careful evaluation demonstrates that they contain a rather subtle constraint that implies that damping is of the classical type. The free vibration elicited by non-vanishing initial conditions is also considered. As a result, we ultimately obtain the general solution for MDOF systems with arbitrary dynamic loads and initial conditions. For undamped systems, the equivalence of the classical modal analysis with the direct approach proposed herein is demonstrated analytically. Special closed-form solutions, singular solutions and numerical examples are also investigated to a limited extent so as to further illustrate the application of the proposed formulae in the context of practical problems.

Random Vibration and Dynamic Reliability Analyses for Nonlinear MDOF Systems under Additive Excitations via DPIM

AbstractIn this paper, a novel direct probability integral method (DPIM) is proposed to address stochastic response and dynamic reliability analyses of nonlinear multidegree-of-freedom (MDOF) syste...