Abstract ：

Multiple tuned mass dampers (MTMD) are widely used to mitigate structural seismic response. With the emergence of more and more high-rise buildings, various MTMD design methods applicable to multi degree of freedom (MDOF) model have been developed. However, the existing MTMD design methods for MDOF models often involve complex optimization algorithms. In order to simplify the MTMD design while effectively reducing the seismic response of the structure, this paper proposes an explicit frequency domain transfer function, theoretically showing the mechanism of decomposing the seismic mitigation of MDOF model into simultaneous seismic mitigation of multiple modes, and developed a theoretical design method of MTMD, so that the dynamic parameters and position of each TMD can be directly obtained based on the developed explicit design formula. A benchmark model was used as an example to verify the seismic mitigation performance. It is shown that the developed multimodal control MTMD design method achieves 22.25% reduction rate of acceleration and 18.5% reduction rate of displacement with 3% mass ratio.

Keyword ：

Seismic mitigation Multiple tuned mass dampers Multi degree of freedom model Frequency domain transfer function Theoretical design method Multimodal control

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Abstract ：

It is well-known that the responses of a structure are different when subjected to a static load or a sudden step load. The dynamic amplification factor (DAF), which is defined as the ratio of the amplitude of the vibratory response to the static response, is normally used to depict the dynamic effect. For a single-degree-of-freedom system (SDOF) subjected to a sudden dynamic load, the maximum value of DAF is 2. Many design guidelines therefore use 2 as an upper bound to consider the dynamic effect. For a civil engineering structure, which is normally a multiple-degrees-of-freedom (MDOF) system, the DAF may exceed 2 in certain circumstances. The adoption of 2 as the upper bond as suggested by the design guidelines therefore may lead to unsafe structural design. Very limited studies systematically investigate the DAF of a MDOF system. This study theoretically investigates the DAF of a MDOF system when it is subjected to a step load based on the fundamental theory of structural dynamics. The condition on which the DAF may exceed 2 is defined. Two numerical examples and one experimental study of a cable-stayed bridge subjected to sudden cable loss are presented to illustrate the problem.

Keyword ：

multiple-degrees-of-freedom (MDOF) step load dynamic amplification factors (DAF)

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Abstract ：

The structural dynamic behavior and yield strength considering both ductility and strain rate effects are analyzed in this article. For the single-degree-of-freedom (SDOF) system, the relationship between the relative velocity and the strain rate response is deduced and the strain rate spectrum is presented. The ductility factor can be incorporated into the strain rate spectrum conveniently based on the constant-ductility velocity response spectrum. With the application of strain rate spectrum, it is convenient to consider the ductility and strain rate effects in engineering practice. The modal combination method, i.e., square root of the sum of the squares (SRSS) method, is employed to calculate the maximum strain rate of the elastoplastic multiple-degree-of-freedom (MDOF) system under uniform excitation. Considering the spatially varying ground motions, a new response spectrum method is developed by incorporating the ductility factor and strain rate into the conventional response spectrum method. In order to further analyze the effects of strain rate and ductility on structural dynamic behavior and yield strength, the cantilever beam (one-dimensional) and the triangular element (two-dimensional) are taken as numerical examples to calculate their seismic responses in time domain. Numerical results show that the permanent displacements with and without considering the strain rate effect are significantly different from each other. It is not only necessary in theory but also significant in engineering practice to take the ductility and strain rate effects into consideration.

Keyword ：

strain rate spectrum multiple-support earthquake excitations ductility effect response spectrum method strain rate effect

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Abstract ：

Basic concept, damping mechanism, application and studying status of particle damping technology were introduced. Its theoretical analysis, numerical simulations and experimental study developments were systematically presented. A discrete element method for simulating a multi-degree-of-freedom (MDOF) structure with a particle damper system was illustrated. The application of particle damping technology in civil engineering and its study situation were mainly discussed, its advantages and potential usages were demonstrated, the difficulties and deficiencies in the present studies were also indicated, and some suggestions were provided.

Keyword ：

Damping Degrees of freedom (mechanics) Finite difference method Gears Ultrasonic devices Vibration control

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Abstract ：

Based on the energy analysis of a multi-degree-of-freedom (MDOF) system, a mode decomposition method for the input seismic energy is proposed, which substitutes a MDOF system with equivalent single-degree-of-freedom (SDOF) systems. The input seismic energy is obtained, using the actual earthquake data. The influence of ground motion parameters is analyzed. The results show that the seismic input energy increases proportionally with the square of ground motion acceleration peak. The bigger the predominant period is, the more the input seismic energy is. © (2012) Trans Tech Publications, Switzerland.

Keyword ：

Acceleration Civil engineering Earthquakes Energy management Fluid mechanics Shore protection Ultrasonic devices

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Abstract ：

In normal circumstances, numerous practical engineering problems are multi-degree-of-freedom (MDOF) nonlinear non-autonomous dynamical systems. Generally, exact solutions for MDOF dynamical systems are hardly obtained; thus, the development of analytical approximations becomes a robust and appealing avenue for an analysis of these systems. The homotopy analysis method (HAM) is one of the analytical methods, which can overcome the foregoing barriers of conventional asymptotic techniques. It has been widely used for solving various nonlinear problems in physical science and engineering. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for MDOF weakly damped non-autonomous dynamical systems. In terms of its flexibility and applicability, the EHAM is also applied to derive the approximate solutions of parametrically and externally excited thin plate systems. Besides, comparisons are performed between the results obtained by the EHAM and the numerical integration (i.e. Runge-Kutta) method. The present findings show that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions.

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Abstract ：

In this study, the extended homotopy analysis method (EHAM) is applied to derive the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled oscillators. The present paper not only introduces the rationale for the EHAM for MDOF oscillators, but also strengthens the availability of the conventional homotopy analysis method (HAM) in solving complex MDOF dynamical systems. Employing the EHAM for the two-degree-of-freedom (TDOF) coupled van der Pol-Duffing oscillator, the explicit analytical solutions of frequency omega and displacements x (1)(t) and x (2)(t) are formulated for various initial conditions and physical parameters. To verify the accuracy and correctness of this approach, a number of comparisons are conducted between the results of the EHAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the third-order analytical solutions of the EHAM agree well with the numerical integration solutions, even if the time variable t progresses to a comparatively large domain in the time history responses.

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Abstract ：

The mathematical models of multi-degree-of-freedom (MDOF) strongly nonlinear dynamical systems are described by coupled second-order differential equations. In general, the exact solutions of MDOF strongly nonlinear dynamical systems are frequently unavailable. Therefore, efforts have been mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic tool for solving strongly nonlinear dynamical systems, and it provides a simple way to ensure the convergence of solution series by means of a convergence-control parameter (h) over bar. Unlike the classical perturbation techniques, this method is independent of the presence of small parameters in the governing equations of motion. In this paper, the HAM is applied to formulate the analytical approximate periodic solutions of MDOF strongly nonlinear coupled van der Pol oscillators. Within this research framework, the frequency and the displacements of two-degree-of-freedom (2-DOF) strongly nonlinear systems can be explicitly obtained. For authentication, comparisons are carried out between the results obtained by the homotopy analysis and numerical integration methods. It is shown that the fourth-order or eighth-order solutions of the present method provide excellent accuracy. Illustrative examples of three-degree-of-freedom (3-DOF) strongly nonlinear coupled van der Pol oscillators are also presented and discussed. Finally, the optimal HAM approach is used to accelerate the convergence of the solutions.

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Abstract ：

In normal circumstances, many practical engineering problems are nonlinear and can be described by multi-degree-of-freedom (MDOF) dynamical systems. Theoretically speaking, the exact solutions are very scarce, so it is extremely significant to develop the analytic tools for nonlinear systems in engineering. Inasmuch as the homotopy analysis method (HAM) can overcome the foregoing restrictions of conventional perturbation techniques, this method has been widely applied to solve a variety of nonlinear problems. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate periodic solutions for MDOF nonlinear dynamic system. The periodic solutions for the parametric excitation buckled thin plate system of MDOF are applied to illustrate the validity and great potential of this method. In addition, comparisons are conducted between the results obtained by the EHAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the second-order analytical solutions of the EHAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

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Abstract ：

Innumerable engineering problems can be described by multi-degree-of-freedom (MDOF) nonlinear dynamical systems. The theoretical modelling of such systems is often governed by a set of coupled second-order differential equations. Albeit that it is extremely difficult to find their exact solutions, the research efforts are mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic technique for solving nonlinear dynamical systems and the method is independent on the presence of small parameters in the governing equations. More importantly, unlike classical perturbation technique, it provides a simple way to ensure the convergence of solution series by means of an auxiliary parameter It. In this paper, the HAM is presented to establish the analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol oscillators. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the higher-order analytical solutions of the HAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

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