Damped Primary Structure Research Articles

Exact H∞ optimization of dynamic vibration absorbers: Univariate-polynomial-based algorithm and operability analysis

H∞ optimization of dynamic vibration absorbers (DVAs) to minimize the maximum response amplitude of primary structures is a classic topic. The commonly used fixed-point method only provides approximate solutions requiring the primary structure is undamped. Instead, we perform exact optimization and investigate the less-reported parametric effects on optimization operability. To handle the known restrictions posed by grounded dampers, a typical DVA model mounted on a damped primary structure, with components connected to both the primary and the base, is considered. We explore three elaborated cases depending on the grounded dampers distributing in the primary structure and the DVA. Our findings reveal that the frequency responses of the primary structure with dual resonant peaks of equal height may not be the global optimum, and we establish a nontrivial necessary condition for operable exact optimization. Furthermore, we elucidate the effects of structural arrangements on the optimized results and provide design rules to maximize vibration suppression performance. The optimization follows the proposal of a so-called resultant-based algorithm that guarantees global optimum with high efficiency and a generalizable core by exclusively constructing univariate polynomial equations. This study contributes a systematic analysis framework alongside efficient calculation tools for exact optimization and DVA performance evaluation.

Parameter optimization of tuned inerter damper for vibration suppression in structures with damping

This study investigates the parameter optimization of tuned inerter damper (TID) in a single degree of freedom (SDOF) system, with a focus on vibration mitigation. As a type of inerter based dynamic vibration absorber (IDVA), TID achieves vibration suppression of the primary structure by replacing the mass block in traditional dynamic vibration absorber (DVA) with an inerter, thereby minimizing the increase in physical mass. With the displacement response of the primary structure as the objective function, this study first employs the classical fixed point theory (FPT) to derive the analytical formulations for the optimal parameters of TID following the H∞ optimization approach and neglecting the inherent damping of primary structure. The influence of optimal parameter deviations on the vibration mitigation effect of TID is also analyzed, revealing that the deviation of the optimal natural frequency ratio has a significant impact on the vibration mitigation performance. By flattening the amplitude curve of the displacement transfer function of primary structure between the fixed points, this study derives the analytical solution for the optimal damping ratio of TID using the extended fixed point theory (EFPT). When the inherent damping of primary structure is considered, this study employs the approximate extended fixed point theory (AEFPT) to derive the approximate optimal parameters of TID. A comparative study of the optimal natural frequency ratios obtained using FPT, AEFPT and numerical searches reveals that the discrepancies among the three methods are minimal for structures with low inherent damping. However, as the inherent damping ratio of the primary structure increases, the optimal natural frequency ratio obtained using FPT deviates significantly from the exact value, whereas the approximate optimal natural frequency ratio derived using AEFPT can significantly reduce this deviation. Combining the conclusion that the vibration mitigation effect of TID is significantly influenced by the natural frequency ratio, this study suggests that for highly damped primary structures, the use of AEFPT can yield more optimal TID parameters, thereby enhancing the robustness of vibration mitigation performance.

Dynamical analysis and numerical verification of a non-smooth nonlinear energy sink

The performance of the vibration reduction of a lightweight non-smooth nonlinear energy sink (NES) attached to a two-story linear damped primary structure (PS) is studied through dynamical analysis and numerical verification. First, a three-degree-of-freedom non-smooth nonlinear system is formulated to describe the coupled dynamics involving the PS and non-smooth NES, in which nonlinearity is induced by the piecewise linear restoring force of the NES and described by a non-smooth function. Second, when a 1:1:1 internal resonance case of the coupled system is considered, a modified complex-average technique and the method of multiple scales are employed to obtain a set of algebraic equations between the vibration amplitudes of the PS and attached non-smooth NES. The stability conditions and initial critical energy for activating targeted energy transfer (TET) can be analytically obtained. Finally, the linear stiffness, equivalent viscous damping coefficients and impact clearance of the non-smooth NES are numerically optimized by the peak estimates. The initial critical energy for the TET of NES is numerically verified to show the excellent performance of vibration reduction for the non-smooth NES by comparing with other attached dissipation devices under a wide range of impulsive loads.

Parameter Optimization and Application for the Inerter-Based Tuned Type Dynamic Vibration Absorbers

As an acceleration-type mechanical element, inerter element has been widely used in the dynamic suppressing field. In this paper, a tuned mass damper with inerter (TMDI) is presented for vibration control and energy dissipation. To evaluate the effectiveness of the TMDI, the simplified model of TMDI coupled with a single-degree-of-freedom (SDOF) structure has been established. Numerical optimization has been conducted with the goal of minimizing the maximum transfer function amplitude of displacement for the damped primary structure. The control performance and robustness for TMDI has been evaluated with the SDOF system in the frequency and time domain, compared with the classical TMD device. Lately, multiple active TMDI (MATMDI) has been proposed as a vibration suppression strategy for a multi-story steel structure. The performances of passive and active control methods have been evaluated in the time domain via real earthquake excitations, and it has proven that the MATMDI is more effective at reducing the response of the structure and the stroke of devices. The results show that the proposed optimal TMDI system can sufficiently harvest vibrational energy and enhance the robustness of structure.

Optimization design for multiple dynamic vibration absorbers on damped structures using equivalent linearization method

The dynamic vibration absorber and tuned mass damper are widely used to suppress harmful vibration of the damped structures under external excitation. The multiple dynamic vibration absorbers have more benefit than the single dynamic vibration absorber. The multiple dynamic vibration absorbers are portability and easy to install because its size is significantly reduced compared to an individual damper. This paper proposes a design method to obtain optimal parameters of multiple dynamic vibration absorbers attached on damped primary structures by using the least squares estimation of equivalent linearization method. An explicit expression of damping ratio and tuning parameters of multiple dynamic vibration absorbers are determined for minimizing the maximum displacement of the primary structures based on the fixed-point theory. The new contribution is provided a reliable theoretical basis for optimizing parameters of the multiple dynamic vibration absorbers that are attached on the damped primary structures. The numerical results reveal the effectiveness of the proposed optimal parameters of multiple dynamic vibration absorbers in reduce vibration of damped primary structures. In the practical applications, this research results allow to divide a large dynamic vibration absorber into many equivalent small dynamic vibration absorbers, which are convenient for manufacturing and installing on the damped primary structures such as high buildings and cable-stayed bridges.

Application of an Artificial Fish Swarm Algorithm in an Optimum Tuned Mass Damper Design for a Pedestrian Bridge

Tuned mass damper (TMD) has a wide application in the human-induced vibration control of pedestrian bridges and its parameters have great influence on the control effects, hence it should be well designed. A new optimization method for a TMD system is proposed in this paper, based on the artificial fish swarm algorithm (AFSA), and the primary structural damping is taken into consideration. The optimization goal is to minimize the maximum dynamic amplification factor of the primary structure under external harmonic excitations. As a result, the optimized TMD has a smaller maximum dynamic amplification factor and better robustness. The optimum TMD parameters for a damped primary structure with different damping ratios and different TMD mass ratios are summarized in a table for simple, practical design, and the fitting equation is also provided. The TMD configuration optimized by the proposed method was shown to be superior to that optimized by other classical optimization methods. Finally, the application of an optimized TMD based on AFSA for a pedestrian bridge is proposed as a case study. The results show that the TMD designed based on AFSA has a smaller maximum dynamic amplification factor than the TMD designed based on the classic Den Hartog method and the TMD designed based on the Ioi Toshihiro method, and the optimized TMD has a good effect in controlling human-induced vibrations at different frequencies.

Weighted averaging technique for the design of dynamic vibration absorber installed in damped primary structures

The dynamic vibration absorber (DVA) has been widely applied in various technical fields. This paper presents a simple approach to determine a closed-form expression for the tuning ratio of a DVA attached to a damped primary structure. The result is obtained by using the so-called weighted averaging technique of the equivalent linearization method proposed by the first author. The values of the tuning ratio given in this paper are compared with those obtained numerically as well as the ones obtained from other authors. The comparison shows the reliability of the method given in this study.

Experimental Study on Vibration Control of a Submerged Pipeline Model by Eddy Current Tuned Mass Damper

Undesirable vibrations occurring in undersea pipeline structures due to ocean currents may shorten the lifecycle of pipeline structures and even lead to their failure. Therefore, it is desirable to find a feasible and effective device to suppress the subsea vibration. Eddy current tuned mass damper (ECTMD), which employs the damping force generated by the relative movement of a non-magnetic conductive metal (such as copper or aluminum) through a magnetic field, is demonstrated to be an efficient way in structural vibration control. However, the feasibility and effectiveness of ECTMD in a seawater environment has not been reported on before. In this paper, an experiment is conducted to validate the feasibility of an eddy current damper in a seawater environment. A submerged pipeline is used as the controlled structure to experimentally study the effectiveness of ECTMD. The dynamic properties of the submerged pipeline are obtained from dynamic tests and the finite element method (FEM). The optimum design of TMD with a linear spring-damper element for a damped primary structure is carried out through numerical optimization procedures and is used to determine the optimal frequency tuning ratio and damping ratio of ECTMD. In addition, the performance of ECTMD to control the submerged pipeline model is respectively studied in free vibration case and forced vibration case. The results show that the damping provided by eddy current in a seawater environment is only slightly varied compared to that in an air environment. With the optimal ECTMD control, vibration response of the submerged pipeline is significantly decreased.

LQG Control of Along-Wind Responses of Tall Building Using Composite Tuned Mass Dampers

A composite tuned mass damper(CTMD) is a vibration control device consisting of an active-passive tuned mass dampers supported on the primary vibrating structure. The performance of CTMD in mitigating wind-induced vibration of tall building is investigated. Optimum parameters of a passive tuned mass damper(PTMD)for minimizing the variance response of the damped primary structure under random loads, with different mass ratio of an active tuned mass damper(ATMD) to a PTMD have been used for the optimum parameters of CTMD. The active control force generated by ATMD actuator was estimated by using linear quadratic Gaussian(LQG) controller, and the fluctuating along-wind load, treated as a stationary random process ,was simulated numerically using the along-wind load spectrum proposed by Solari .Comparing the along-wind rms response of tall building without a CTMD, the CTMD is effective in reducing the response to 40%~45% of the response without the CTMD. Therefore, the CTMD system was effective in reducing wind-induced vibration of tall building.

MULTIPLE TUNED MASS DAMPERS FOR SEISMIC-EXCITED TALL BUILDING

Multiple tuned mass dampers (MTMD) is used for suppressing seismic excitation of tall building. The performance of MTMD is compared with that of a single tuned mass system (TMD). Optimum parameters of TMD/MTMD for minimizing the variance response of the damped primary structure derived by Krenk, Igusa, and Jangid were used. The optimally designed MTMD system is more effective than that of a single TMD system for reducing seismic excitation of a tall building.

Research on the design of non-traditional dynamic vibration absorber for damped structures under ground motion

An analytical approach is presented to investigate the optimal problem of non-traditional type of Dynamic vibration absorber (DVA) for damped primary structures subjected to ground motion. Different from the standard configuration, the non-traditional DVA contains a linear viscous damper connecting the absorber mass directly to the ground instead of the main mass. There have been many studies on the design of the non-traditional DVA for undamped primary structures. Those studies have shown that the non-traditional DVA produces better performance than the standard DVA does. When damping is present at the primary system, there are very few works on the non-traditional dynamic vibration absorber. To the best of our knowledge, there is no study on the design of non-traditional DVA for damped structures under ground motion. We propose a simple method to determine the approximate analytical solutions of the nontraditional DVA when the damped primary structure is subjected to ground motion. The main idea of the study is based on the criterion of the equivalent linearization method to replace approximately the original damped structure by an equivalent undamped one. Then the approximate analytical solution of the DVA’s parameters is given by using known results for the undamped structure obtained. Comparisons have been done to validate the effectiveness of the obtained results.

Tuned vibration absorbers with nonlinear viscous damping for damped structures under random load

The classical problem for the application of a tuned vibration absorber is to minimize the response of a structural system, such as displacement, velocity, acceleration or to maximize the energy dissipated by tuned vibration absorber. The development of explicit optimal absorber parameters is challenging for a damped structural system since the fixed points no longer exist in the frequency response curve. This paper aims at deriving a set of simple design formula of tuned vibration absorber with nonlinear viscous damping based on the frequency tuning for harmonic load for a damped structural system under white noise excitation. The vibration absorbers being considered include tuned mass damper (TMD) and liquid column vibration absorber (LCVA). Simple approximate expression for the standard deviation velocity response of tuned vibration absorber for damped primary structure is also derived in this study to facilitate the estimation of the damping coefficient of TMD with nonlinear viscous damping and the head loss coefficient of LCVA. The derived results indicate that the higher the structural inherent damping the smaller the supplementary damping provided by a tuned vibration absorber. Furthermore, the optimal damping of tuned vibration absorber is shown to be independent of structural damping when it is tuned using the frequency tuning for harmonic load. Finally, the derived closed-form expressions are demonstrated to be capable of predicting the optimal parameters of tuned vibration absorbers with sufficient accuracy for preliminary design of tuned vibration absorbers with nonlinear viscous damping for a damped primary structure.

Global-local approach to the design of dynamic vibration absorber for damped structures

The dynamic vibration absorber (DVA) has attracted attention since its invention. This paper deals with the optimization problems of the standard DVA and two other models of DVA called three-element DVA and non-traditional DVA for damped primary structures. Unlike the standard configuration, the three-element DVA contains two spring elements in which one is connected to a dashpot in a series and the other is placed in parallel. Meanwhile the non-traditional DVA has a linear viscous damper connecting the absorber mass directly to the ground. There have been some studies on the design of three-element and non-traditional dynamic vibration absorbers in the case of undamped primary structures. These studies have shown that both three-element and non-traditional DVAs perform better than the standard DVA. When the primary structure is damped, there are very few studies on the three-element and non-traditional DVAs in the literature. This article proposes a global-local approach to give approximate analytical solutions of the optimization for all standard, three-element and non-traditional DVAs attached to damped primary structures. The main idea of the study is based on the global-local criterion of the equivalent linearization method in order to replace approximately the original damped structure by an equivalent undamped one. Afterwards, the already derived expressions of the optimal parameters for the undamped primary system case are used with the equivalent undamped structure that was just obtained. The numerical simulations are carried out to verify the effectiveness of the obtained results. Additionally, design aids, which show the variation of the optimal design quantities for various DVA mass ratios and inherent structural damping ratios are also provided.

Optimal design of a novel tuned mass-damper–inerter (TMDI) passive vibration control configuration for stochastically support-excited structural systems

This paper proposes a novel passive vibration control configuration, namely the tuned mass-damper–inerter (TMDI), introduced as a generalization of the classical tuned mass-damper (TMD), to suppress the oscillatory motion of stochastically support excited mechanical cascaded (chain-like) systems. The TMDI takes advantage of the “mass amplification effect” of the inerter, a two-terminal flywheel device developing resisting forces proportional to the relative acceleration of its terminals, to achieve enhanced performance compared to the classical TMD. Specifically, it is analytically shown that optimally designed TMDI outperforms the classical TMD in minimizing the displacement variance of undamped single-degree-of-freedom (SDOF) white-noise excited primary structures. For this particular case, optimal TMDI parameters are derived in closed-form as functions of the TMD mass and the inerter constant. Furthermore, pertinent numerical data are furnished, derived by means of a numerical optimization procedure, for a 3-DOF classically damped primary structure base excited by stationary colored noise, which exemplify the effectiveness of the TMDI over the classical TMD to suppress the fundamental mode of vibration for MDOF structures. It is concluded that the incorporation of the inerter in the proposed TMDI configuration can either replace part of the TMD vibrating mass to achieve lightweight passive vibration control solutions, or improve the performance of the classical TMD for a given TMD mass.

Design of three-element dynamic vibration absorber for damped linear structures

The standard type of dynamic vibration absorber (DVA) called the Voigt DVA is a classical model and has long been investigated. In the paper, we will consider an optimization problem of another model of DVA that is called three-element type DVA for damped primary structures. Unlike the standard absorber configuration, the three-element DVA contains two spring elements in which one is connected to a dashpot in series and the other is placed in parallel. There have been some studies on the design of the three-element DVA for undamped primary structures. Those studies have shown that the three-element DVA produces better performance than the Voigt DVA does. When damping is present at the primary system, to the best knowledge of the authors, there has been no study on the three-element dynamic vibration absorber. This work presents a simple approach to determine the approximate analytical solutions for the H∞ optimization of the three-element DVA attached to the damped primary structure. The main idea of the study is based on the criteria of the equivalent linearization method in order to replace approximately the original damped structure by an equivalent undamped one. Then the approximate analytical solution of the DVA's parameters is given by using known results for the undamped structure obtained. The comparisons have been done to verify the effectiveness of the obtained results.

Design of non-traditional dynamic vibration absorber for damped linear structures

The Voigt-type of dynamic vibration absorber is a classical model and has attracted considerable attention in many years because of its simple design, high reliability and useful applications in the fields of civil and mechanical engineering. Recently, a non-traditional type of dynamic vibration absorber was proposed. Unlike the traditional damped absorber configuration, the non-traditional absorber has a linear viscous damper connecting the absorber mass directly to the ground instead of the main mass. There have been some studies on the design of the non-traditional dynamic vibration absorber in the case of undamped primary structures. Those studies have shown that the non-traditional dynamic vibration absorber has better performance than the traditional dynamic vibration absorber. However, when damping is present at the primary system, there are very few studies on the design of non-traditional dynamic vibration absorber. This article presents a simple approach to determine the approximate analytical solutions for the [Formula: see text] optimization of the non-traditional dynamic vibration absorber attached to the damped primary structure subjected to force excitation. The main idea of the study is based on the dual criterion suggested by Anh in order to replace approximately the original damped structure by an equivalent undamped structure. Then the approximate analytical solution of dynamic vibration absorber’s parameters is given by using known results for undamped structure obtained. The comparisons have been done to verify the effectiveness of the obtained results.

Closed-Form Optimum Liquid Column Vibration Absorber Parameters for Base-Excited Damped Structures

There have been significant development of techniques for determining optimum parameters of passive dampers and vibration absorbers, but most of the existing studies are limited to the case of an undamped primary structure. This paper aims to develop a first order approximation of the optimal liquid column vibration absorber (LCVA) parameters for suppressing the displacement response of a damped structure under white noise induced ground acceleration. The optimum parameters of a LCVA for the case of an undamped primary structure subjected to white noise induced ground acceleration are first derived and the results are then extended to the case where some damping exists in the primary structure by using a perturbation technique. The accuracy of the derived formulas is then verified through a comparison with the results obtained from numerical methods as well as the results of other studies found in the literature. Finally, a direct approach for determining the optimal head loss coefficient is presented in this study. It is demonstrated that the closed-form approach is capable of predicting the optimal liquid column vibration absorber parameters with reasonable accuracy for damped primary structure.

Closed form optimal solution of a tuned liquid column damper for suppressing harmonic vibration of structures

Building vibration has become one of the major concerns to structural engineers. Being a vibration absorber, a tuned liquid column damper (TLCD) not only suppresses building vibration, but also acts as the water storage facility of a building. In view of this, the application of such damper is particularly suited for building structures. However, due to inherent nonlinear liquid damping, it is no doubt that a great deal of computational effort is required to search the optimum parameters of TLCD numerically. Therefore, this paper aims to develop a closed form solution scheme of TLCD-structure systems and the optimum parameters of TLCD that lead to the maximum vibration reduction of building structure, so as to facilitate the design of dampers. The developed closed form solution of TLCD-structure system is verified by comparing results obtained from the conventional iterative method. After having a satisfactory verification, the existence of the invariant points of a TLCD-structure system for the case of undamped primary structure is demonstrated. Explicit design formulas of TLCD for the case of undamped primary structures are obtained by optimizing the response at the two invariant points. The optimum parameters of TLCD for the case of damped primary structures are also investigated numerically. An approach for determining the optimum head loss coefficient of TLCD is proposed. Finally, an example is given to illustrate the design procedures of TLCD for suppressing harmonic type vibrations

Analytical Solutions for DVA Optimization Based on the Lyapunov Equation

A novel method is proposed to obtain the optimum configuration of dynamic vibration absorber (i.e., DVA) attached to an undamped or damped primary structure. The performance index, i.e., the quadratic integration, includes two types of controls, i.e., velocity and displacement controls of the primary mass. Based on the Lyapunov equation, the performance indices are simplified into matrix quadratic forms. With the help of the Kronecker product and matrix column expansion, the closed-form solutions of optimum parameters for undamped primary structure are finally presented. Moreover, in some cases, the method can produce perturbation solutions in simple forms for damped primary structure. Especially, from these solutions, the classical optimum H2 designs under external force or base acceleration excitation can be derived out.

Perturbed complex eigenproperties of classically damped primary structure and equipment systems

In the dynamic analysis of light equipment supported on structures, it sometimes becomes essential to consider the effect of the non-classicality of damping on the dynamic characteristics and the equipment response, even if the supporting structure is classically damped. This non-classical damping effect can be included in dynamic analysis through a state vector approach wherein damped and complex-valued eigenproperties are used. Herein a perturbation approach is developed to obtain such damped eigenproperties of the combined structure-equipment system in terms of the undamped eigenproperties of the supporting structure and the equipment. The closed form expressions are obtained for the eigenvalues and eigenvectors of both the detuned and tuned equipment. These can be used to obtain the dynamic response of equipment, such as seismic floor response spectra, which incorporate the non-classical damping effects and dynamic interaction between the two systems. The numerical results demonstrating the applicability of the proposed approach are presented.