Equivalent Linear Damping Coefficient Research Articles

Simplified damping model of the eddy current damper with a double-annular-plate configuration

The eddy current damper has plenty of advantages, such as non-contact energy dissipation, stable damping performance, and high reliability and durability, which is a novel damping device with great potential to apply in various fields. However, there are still many challenges in the applications of eddy current dampers for vibration suppression, especially in confined installation environments of aerospace structures, such as the insufficient energy dissipation efficiency of miniaturized eddy current dampers and the nonlinear characterization of eddy current damping under high frequency conditions. This study aims to propose a new design scheme for the eddy current damper to achieve high energy dissipation efficiency from lightweight and miniaturized eddy current dampers. By comparing with the composite planar configuration (which provides the highest energy dissipation efficiency in the literature) designed by Chen's group through the finite element analysis, the proposed double-annular-plate configuration achieves much better damping performance in the miniaturized version for the applications in confined installation spaces. To help the future design of the damper and uncover the underlying principle resulting in the nonlinear effect, based on the generation mechanism of eddy current damping, a simplified damping model for the double-annular-plate eddy current damper is established considering the nonlinearity of eddy current damping at high frequencies. The simplified model shows that the nonlinearity of eddy current damping mainly depends on the phase delay between the damping force and the excitation displacement. Moreover, based on the time-varying electromagnetic field theory, a mechanical-magnetic coupling finite element model is established to simulate the damping performance of the proposed damper. After validating the simplified damping model by the finite element analysis, a prototype of the double-annular-plate eddy current damper is manufactured and its dynamic behaviors and damping characteristics are measured by experiments. The results show that there is good agreement among the calculation results by the simplified model, the calculation results by the finite element analysis simulation, and the experiment results. The damping force amplitude of the double-annular-plate damper is approximately proportional to the frequency, and the delayed initial phase is slightly less than -π/2 and decreases with the increase of the frequency. The equivalent linear damping coefficient of the damper also decreases with the increase of the frequency, which shows the nonlinear effect of eddy current damping. Therefore, the proposed simplified theoretical model and the finite element model can efficiently characterize the nonlinearity of eddy current damping and can be well utilized for the mechanical and damping characteristics analysis and optimization design of eddy current dampers, which provides theoretical reference and technical support for the application of the novel damper in the aerospace field.

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.

Equivalent linear roll damping of a FPSO coupled with liquid sloshing in a pair of two-row tanks

The increased risk of reduced transverse stability by free surfaces in cargo tanks in floating vessels such as Floating Production Storage and Offloading (FPSO) units, is well known. Roll decay testing remains the most common method for determining roll damping characteristics. Few works evaluating roll characteristics coupled with the effect of liquid motion exist. In this paper, a series of roll decay tests are carried out for a FPSO model with a pair of two-row prismatic tanks. The four compartment tanks are filled to different levels, and the equivalent linear damping is obtained using the Quasilinear (Logarithmic Decrement) method and the Froude Energy method. The effect of draft change for each loading condition is analyzed based on the average equivalent linear damping coefficient and the effect of the liquid cargo sloshing is also discussed. The results show that an increase in liquid cargo, the location of cargo load, and the geometry of the cargo tank affect the damping of the vessel.

Dynamical response of fractional-order nonlinear system with combined parametric and forcing excitation

A dynamical analysis of a Mathieu-van der Pol-Duffing nonlinear system with fractional-order derivative under combined parametric and forcing excitation is studied in this paper. The approximate analytical solution is researched for 1/2 sub harmonic resonance coupled with primary parametric resonance based on the improved averaging approach. The steady-state periodic solution including its stability condition is established. The equivalent linear stiffness coefficient (ELDC) and equivalent linear damping coefficient (ELSC) for this nonlinear fractional-order oscillator are defined. Then, the numerical simulations are presented in three typical cases by iterative algorithms. The time history, phase portrait, FFT spectrum and Poincare maps are shown to explain the system response. Some different responses, such as quasi-periodic, multi-periodic and single periodic behaviors are observed and investigated. The results of comparisons between the numerical solutions and the approximate analytical solutions in three typical cases show the correctness of the analytical solutions. The influences of the fractional-order parameters on the system dynamical response are researched based on the ELDC and ELSC. Through analysis, it could be found that the increase of the fractional-order coefficient would result in the rightward and downward movements of the amplitude-frequency curves. The increase of the fractional-order coefficient will also move the bifurcation point rightwards and will make the existing range of steady-state solution larger. It could also be found that the ELSC will become larger and ELDC smaller when the fractional order is closer to zero, so that the decrease of the fractional order would make the response amplitude larger. At last, the detailed conclusions are summarized, which is beneficial to design and control this kind of fractional-order nonlinear system.

A novel approach for the design and analysis of nonlinear dampers for automotive suspensions

This paper proposes an analytical technique for frequency analysis and the design of nonlinear dampers to further improve ride dynamics performance of vehicle suspensions over a wide range of excitation frequencies. Using the energy balance method (EBM), the proposed methodology estimates the equivalent linear damping coefficient of any nonlinear passive damper whose force is a general function of the damper’s relative displacement and relative velocity. Knowing the equivalent linear damping coefficient makes it possible to perform a frequency analysis of the suspension ride performance with any nonlinear damper. Some specific criteria are defined to design the desired form of equivalent linear damping coefficient which provides a high/small damping ratio at low-/high-frequency excitations, so the corresponding nonlinear damping force required to obtain improved ride performance of the suspension using a 1-degree-of-freedom quarter car model is also defined. A sensitivity analysis is then performed to provide a design guideline. The results show that the dependency of the equivalent damping coefficient either relative to the velocity of the suspension (velocity-dependent damper) or the relative displacement of the suspension (position-dependent damper) could provide a variable damping ratio leading to better vibration isolation over the excitation frequency. A noticeable ride dynamic performance can be reached over the entire range of the excitation frequency by designing a nonlinear damper such that its equivalent linear damping ratio becomes a desired function of both its relative displacement and relative velocity (position-velocity-dependent damper).

Equivalent linear damping characterization in linear and nonlinear force-stiffness muscle models.

In the current research, the muscle equivalent linear damping coefficient which is introduced as the force-velocity relation in a muscle model and the corresponding time constant are investigated. In order to reach this goal, a 1D skeletal muscle model was used. Two characterizations of this model using a linear force-stiffness relationship (Hill-type model) and a nonlinear one have been implemented. The OpenSim platform was used for verification of the model. The isometric activation has been used for the simulation. The equivalent linear damping and the time constant of each model were extracted by using the results obtained from the simulation. The results provide a better insight into the characteristics of each model. It is found that the nonlinear models had a response rate closer to the reality compared to the Hill-type models.

Optimal design for fractional-order active isolation system

The optimal control of fractional-order active isolation system is researched based on the optimal control theory, and the effect of fractional-order derivative on passive isolation system is also analyzed. The mechanical model is established where viscoelastic features of isolation materials are described by fractional-order derivative. The viscoelastic property of the fractional-order derivative in dynamical system is studied and the fractional-order derivative could be divided into linear stiffness and linear damping. It is found that both the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient but also the resonance frequency by the equivalent linear stiffness. Based on optimal control theory, the feedback gain of fractional-order active isolation system under harmonic excitation is obtained, which is changed with the excitation frequency. The statistical responses of the displacement and velocity for passive and active vibration isolation systems subjected to random excitation are also presented, which further verifies the excellent performance of fractional-order derivative in vibration control engineering.

Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative

The subharmonic resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC) for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR) is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.

The Calculation of the Equivalent Linear Damping Coefficient of the Magnetorheological Damper

MR damper (magnetorheological damper) has broad application prospects, and equivalent damping coefficient is very important of its dynamic characteristic analysis. Based on the modified Bouc_Wen model, the performance of MR damper was analyzed and the equivalent linear damping coefficient of MR damper was calculated. Based on simulation date of the modified Bouc_Wen model, the relationships between the equivalent linear damping coefficient of MR damper and the parameters of control voltage and MR dampers movement amplitude were established by the curve fitting regression analysis method. Verification results prove that the equivalent linear damping coefficient model has higher accuracy. For the vibration systems using strongly nonlinear MR damper, new model can effectively improve the efficiency of calculating the vibration analysis and the stability of the system in a certain frequency. At the same time, the model provides a theoretical basis for the application of MR damper control.

Displacement Transmissibility Characteristics of Harmonically Base Excited Damper Isolators with Mixed Viscous Damping

The viscous damping force in the mixed form as fd (˙x )= c1 ˙ x + c2 |˙ x| ˙ x can well describe damping characteristics of isolators and dampers in many cases. In this paper, performance characteristics of single-degree-of-freedom (SDOF) linear- stiffness isolators with mixed and piecewise mixed viscous damping are analytically examined under harmonic base excitation. Based on the first-order harmonic balance method (HBM), both relative and absolute displacement transmissibility expressions with the equivalent linear damping coefficient (ELDC) are given. And the analytical calculations show good agreement with the numerical results. Also, the influence of nonlinear damping on the response characteristics is investigated by comparing the transmissibility of linear and nonlinear systems. The resonant frequency always shifts to a lower value as the nonlinear damping component of the force fd (˙x )= c1 ˙ x + c2 |˙ x| ˙ x becomes stronger, and when the damping ratio in the corresponding linear model is relatively high, the relative transmissibility decreases at frequencies higher than the resonance frequency of the corresponding linear damping system and the absolute one increases for the frequency ratios above √ 2. Finally, the displacement transmissibility of a nonlinear isolator with piecewise mixed viscous damping is discussed and the process shows research similarity with the non-piecewise case.

Displacement Transmissibility Characteristics of Harmonically Base Excited Damper Isolators with Mixed Viscous Damping

The viscous damping force in the mixed form asfd(x˙)=c1x˙+c2|x˙|x˙can well describe damping characteristics of isolators and dampers in many cases. In this paper, performance characteristics of single-degree-of-freedom (SDOF) linear-stiffness isolators with mixed and piecewise mixed viscous damping are analytically examined under harmonic base excitation. Based on the first-order harmonic balance method (HBM), both relative and absolute displacement transmissibility expressions with the equivalent linear damping coefficient (ELDC) are given. And the analytical calculations show good agreement with the numerical results. Also, the influence of nonlinear damping on the response characteristics is investigated by comparing the transmissibility of linear and nonlinear systems. The resonant frequency always shifts to a lower value as the nonlinear damping component of the forcefd(x˙)=c1x˙+c2|x˙|x˙becomes stronger, and when the damping ratio in the corresponding linear model is relatively high, the relative transmissibility decreases at frequencies higher than the resonance frequency of the corresponding linear damping system and the absolute one increases for the frequency ratios above2. Finally, the displacement transmissibility of a nonlinear isolator with piecewise mixed viscous damping is discussed and the process shows research similarity with the non-piecewise case.

Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative

A linear single degree-of-freedom oscillator with fractional-order derivative is researched by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical property, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and this conclusion is entirely different from the published results. The comparison of the analytical solution with the numerical results verifies the correctness of the approximately analytical results. The following analysis on the effects of the fractional parameters on the amplitude-frequency is fulfilled, and it is found that the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness.

Dynamical analysis of linear SDOF oscillator with fractional-order derivative (Ⅱ)

A linear single degree-of-freedom (SDOF) oscillator with two kinds of fractional-order derivatives is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical properties, including the fractional coefficients and the fractional orders in the two kinds of fractional-order derivatives, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and the results is entirely different from the results given in the existing literature. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The following analysis of the effects of the fractional parameters on the amplitude-frequency is presented, and it is found that the fractional coefficients and the fractional orders can affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness. Finally, the effects of the fractional coefficient in the second fractional-order derivative on resonance frequency are analyzed, and the design rule for the fractional coefficient in the second fractional-order derivative to meet the satisfactory vibration control performance is pointed out.

VIBRATION OF CIRCULAR BLADED DISK WITH IMPERFECTIONS

The steady state response of a model of circular bladed disk with imperfection is investigated. Disk imperfection results from additional two groups of damping heads fixed on opposite ends of one diameter. These damping heads are introduced into the computing model as additional point mass, damping and stiffness. Such type of imperfection causes the bifurcation of double eigenfrequencies into pairs of close eigenfrequencies. The effect of imperfection is examined both numerically on three-dimensional nonrotating FE-model and analytically on a simplified split 2DOF model of rotating disk excited by single point harmonic force. Nonlinear friction connection is analyzed and equivalent linear damping coefficient is derived and used in the calculation procedure. It is shown that nonproportional distribution of damping strongly influences the high of resonance peaks. Some examples of response curves illustrate the dynamic properties of stationary and rotating disks with mass-damping-stiffness imperfection.

Influence of excitation amplitude on the characteristics of nonlinear butyl rubber isolators

The purpose of this study is to explore the advantages and characteristics of nonlinear butyl rubber (type IIR) isolators in vibratory shear by comparison with linear isolators. It is known that the mechanical properties of viscoelastic materials exhibit significant frequency and temperature dependence, and in some cases, nonlinear dynamic behavior as well. Nonlinear characteristics in shear deformation are reflected in mechanical properties such as stiffness and damping. Furthermore, even when the excitation amplitude is small the response amplitude may often be large enough that nonlinearities cannot be ignored. The treatment involves developing phenomenological models of the effective storage modulus and effective loss factor of a rubber isolator material as a function of excitation amplitude. The transmissibility of a nonlinear viscoelastic isolator is compared with that of a linear isolator using an equivalent linear damping coefficient. Forced resonance vibration and impedance tests are used to characterize nonlinear parameters and to measure the normalized transmissibility. It is found that as the excitation amplitude of the nonlinear viscoelastic isolator increases, the response amplitude decreases and the transmissibility is improved over that of the linear isolator for excitation frequency that exceeds a particular value governed by the temperature and excitation amplitude. The method of multiple scales and numerical simulations are used to predict the response characteristics of the isolator based on the phenomenological modeling under different values of system parameters.

Damping Of Pedestrian‐Induced Bridge Vibrations By Tuned Liquid Column Dampers

AbstractThe excessive lateral vibrations of Londons Millenium Bridge and the Toda Park Bridge in Japan due to a large number of crossing pedestrians have raised an unexpected problem in footbridge constructions. Secondary tuned structures, like the conventional tuned mass damper (TMD) or the tuned liquid damper (TLD) were installed to the bridge in order to suppress these vibrations. In the present investigation it is proposed to apply the more efficient and more economic tuned liquid column damper (TLCD), which relies on the motion of a liquid mass in a sealed tube to counteract the external motion, while a built‐in orifice plate induces turbulent damping forces that dissipate kinetic energy. For optimal tuning of TLCDs the natural frequency and equivalent linear damping coefficient have to be chosen suitable, likewise to the conventional TMDs, as given in Den Hartog [1]. The advantages of TLCDs are: simple tuning of natural frequency and damping, low cost of design and maintenance and a simple construction. A mathematical model of a three degree‐of‐freedom (DOF) bridge coupled with an optimal tuned TLCD is derived and analyzed numerically. Furthermore, a small scale experimental model set‐up has been constructed in the laboratory of the TU‐Insitute. The experimental results are in good agreement with the theoretical predictions and indicate that TLCDs are effective damping devices for the undesired pedestrian induced footbridge vibrations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis on Wave Transformations and Wave Forces about an Angular Body Considering Wave Energy Dissipations

Concerning wave energy dissipations caused by flow separations and resultant vortex formations from sharp edges of an angular body, a twodimensional analysis on wave transformations and wave forces on an angular body has been developed. In the analysis, a damping wave model was newly proposed to account for the wave energy dissipation approximately. An equivalent linear damping coefficient of fluid and a spatial range of the damping fluid are necessary in the analysis. In order to figure out these factors, wave tank tests were carried out. Two different structural models, i.e., a curtain-walled breakwater and a semi-submerged rectangular body, were used in the experiment. It was confirmed that the numerical analysis developed here is useful for estimating wave transformations and wave forces about an angular body including the effects of wave energy dissipation due to vortex formations around the body.

Shock-Absorbing Mechanism of Human Leg.

The shock-absorbing mechanism to minimize the vertical acceleration when jumping down onto a hard surface was investigated. The dynamic model of a human leg was built with a link joint. The muscle model consisted of a contractile element, a elastic element and a viscous element. The equivalent linear spring constant and equivalent linear damping coefficient decreased upon lowering of the hip by varying the bending angles of the ankle and knee. The numerical results showed that shock could be absorbed passively by deep bending, and to a greater degree actively in the same bending position.

Experimental investigation of the lift component of roll damping

The lift force and turning moment acting on a model towed obliquely to the direction of motion have been measured. Two models were used; one of them was tested fitted with and without a rudder. These measurements were used to determine the magnitude of the lift coefficient and the point of application of the transverse force acting on the model. The data were then used to determine the lift component of the roll damping moment. It has been found that the equivalent linear damping coefficient due to lift is a nonlinear function of the forward speed of the ship.

Mechanical Model for Oscillating Water Column with Compressibility

Certain aspects of modeling of the oscillating water column (OWC) wave‐power device are examined, with particular reference to its behavior at representative off‐resonant frequencies. Although comparison is made with a particular study incorporating the use of compressible flow equations for the air column, this work uses a simple mechanical analog that dispenses with the thermodynamic equations and arrives at similar results. Bearing in mind that the compressible flow study was based on quadratic damping and this one is based on linear damping, the differences when the two results are compared on an energy basis are remarkably small. The compression is made possible by the calculation of an equivalent linear damping coefficient, for the model developed here, from the quadratic damping used elsewhere. The details of the conversion are also provided. For certain studies, it is therefore proposed that the mechanical analog may be used to simplify programming and facilitate evaluation.