Stochastic seismic analysis of structures with nonlinear eddy current dampers

Abstract

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.

Keywords

nonlinear eddy current damper stochastic linearization technique Monte Carlo simulation passive control vibration suppression

Introduction

Passive control with supplemental energy dissipation devices has attracted considerable attention over the past few decades. It has been proven that passive control is very effective in improving the seismic performance of civil structures.^1,2 These energy dissipation devices absorb a large proportion of energy induced by earthquakes, which contributes to mitigating structural damage and reducing retrofitting costs. A variety of energy dissipation devices have been proposed, such as fluid viscous dampers³ (FVD), metallic dampers,⁴ viscoelastic dampers,⁵ and tuned vibration dampers.⁶ The eddy current damper^7–10 (ECD) is a type of novel energy-dissipating devices whose resistant force is generated by interaction between eddy currents and magnetic field. Due to its mechanical contactless feature which avoids the disadvantages of failure risks in FVDs and thus results in long-term stability, ECD has been widely applied in brake systems and vehicle suspension systems.¹¹ However, ECD’s low energy-dissipating density leads to the small feasibility of being applied to control vibration in civil engineering, especially in earthquake protection. Combination with tuned mass or inertial mass¹² is the main method leading ECD into application so far. Efforts have been made to enhance the vibration control effect of ECD, such as optimal design,¹⁰ coupling with another solid energy dissipation mechanism,¹³ and adopting electro-magnetic technique and hybrid control strategy.^14,15

Utilizing the motion magnification mechanisms, like ball screw¹⁶ or rack and gear,¹⁷ to increase input velocity of ECD is another available method to improve eddy current damping effect. But installing these motion magnification mechanisms will change the nonlinearity of the relationship between eddy current damping force and velocity. As the result, the difficulties in the design of the damper devices and the evaluation of the structural response become insurmountable, which also raises in FVD.¹⁸ Therefore, the attempts have been made on linearization for the nonlinear damping force.^19–21 Evaluating the equivalent viscous damping ratio contributed by the nonlinear dampers is the most effective way to assess their energy-dissipating capacity, and energy equivalent method,¹⁹ in one cycle of vibration, is widely adopted in the study of vibration control. Nevertheless, this method based on the hypothetical harmonic analysis does not give the exact estimation of linearized coefficient due to the uncertainties and randomness of earthquake, while the stochastic linearization technique²² (SLT) is an alternative one. The SLT has been employed in a series of researches and exhibited an extraordinary applicability, such as nonlinear viscous dampers²¹ characterized by exponential function, nonlinear hysteretic dampers,²³ and optimal design of nonlinear energy sinks,^24,25 among others.

The purpose of this work is to evaluate the linearization of the nonlinear eddy current damping force under stochastic earthquake characterized by Kanai-Tajimi power spectral density (PSD) via using the stochastic linearization technique. The force-based criterion and energy-based criterion are chosen for linearization. The analysis results via SLT are validated by the Monte Carlo simulations (MCSs) approach performed on the originally nonlinear system.

SDOF structures with nonlinear eddy current dampers

Different from the well-known fluid viscous dampers, the damping force of eddy current dampers increases with velocity before it reaches the critical velocity, and then decreases at a higher velocity input. Its nonlinear force-velocity relationship is characterized precisely by the Wouterse’s model²⁶ written as

\begin{matrix} F_{d} = \frac{2 F_{d m a x}}{\frac{V_{c r}}{| \dot{u} |} + \frac{| \dot{u} |}{V_{c r}}} s g n (\dot{u}) \end{matrix}

(1)

where

F_{d m a x}

is the maximum damping force and

V_{c r}

is the critical velocity which corresponds to the maximum damping force. These damping parameters depending on the constructions of the eddy current dampers are commonly fitted by the prototype experiments and the finite element mothed (FEM) simulations.

The seismic performance of a single degree of freedom system attached with eddy current damper is analyzed under stochastic ground motion that is a stationary zero-mean Gaussian random process described by the PSD function. Figure 1 displays a general sketch of a single degree of freedom system with eddy current damper. The dynamic equation of this system subjected to the ground excitation ${\ddot{u}}_{g}$ is

\begin{matrix} m \ddot{u} + c \dot{u} + k u + F_{d} = - m {\ddot{u}}_{g} \end{matrix}

(2)

where m, c, and k represent the mass, inherent damping coefficient, and stiffness of the system, respectively.

Figure 1.

SDOF building model excited by ground acceleration.

Assume the ground acceleration ${\ddot{u}}_{g}$ is a stationary zero-mean Gaussian random process that can be depicted by the PSD function $S_{{\ddot{u}}_{g}} (ω)$ . The widely known Kanai-Tajimi filtered Gaussian white-noise process²⁷ is developed to describe the earthquake frequency content with two filters decided by the soil characteristics. The two-sided stationary PSD function is

\begin{matrix} S_{{\ddot{u}}_{g}} (ω) = \frac{ω^{4} + 4 ζ_{g}^{2} ω_{g}^{2} ω^{2}}{{(ω_{g}^{2} - ω^{2})}^{2} + 4 ζ_{g}^{2} ω_{g}^{2} ω^{2}} \frac{ω^{4}}{{(ω_{f}^{2} - ω^{2})}^{2} + 4 ζ_{f}^{2} ω_{f}^{2} ω^{2}} S_{w} \end{matrix}

(3)

where

ω_{g}

ζ_{g}

ω_{f}

, and

ζ_{f}

are filter parameters. And the white-noise spectral level

S_{w}

related to the peak ground acceleration (PGA)

{\ddot{u}}_{g 0}

\begin{matrix} S_{w} = \frac{0.141 ζ_{g} {\ddot{u}}_{g 0}^{2}}{ω_{g} \sqrt{1 + 4 ζ_{g}^{2}}} \end{matrix}

(4)

For frequency domain calculation, replacing the nonlinear eddy current damping force by an equivalent one $F_{d}^{l} = c_{d, e q} \dot{u}$ in equation (2) yields

\begin{matrix} m \ddot{u} + (c + c_{d, e q}) \dot{u} + k u = - m {\ddot{u}}_{g} \end{matrix}

(5)

where

c_{d, e q}

denotes the equivalent viscous coefficient that can be calculated by different equivalence criteria of which the force criterion is the most common one. The error or difference between the nonlinear and linearized force is

\begin{matrix} e = F_{d} - F_{d}^{l} = \frac{2 F_{d m a x}}{\frac{V_{c r}}{| \dot{u} |} + \frac{| \dot{u} |}{V_{c r}}} s g n (\dot{u}) - c_{d, e q} \dot{u} \end{matrix}

(6)

To minimize the error in the least square sense, one has the force-based linearization coefficient $c_{d, e q}^{F B}$ as follows

\begin{matrix} \frac{\partial}{\partial c_{d, e q}} E [e^{2}] = 0 \to c_{d, e q}^{F B} = 2 F_{d m a x} V_{c r} \frac{E [{(1 + \frac{V_{c r}^{2}}{{\dot{u}}^{2}})}^{- 1}]}{E [{\dot{u}}^{2}]} \end{matrix}

(7)

Here $E [\cdot]$ denotes the expectation operator, and the superscript FB stands for “force-based” equivalence criterion. One alternative equivalence criterion was proposed by Elishakoff and Zhang²⁸ based on the dissipated energy of nonlinear and linearized force. According to this equivalence criterion, the functions representing the energy dissipated in the nonlinear and linearized systems are

\begin{matrix} ε_{d}^{N L} = \int_{0}^{\dot{u}} \frac{2 F_{d m a x}}{\frac{V_{c r}}{\dot{x}} + \frac{\dot{x}}{V_{c r}}} d \dot{x} = F_{d m a x} V_{c r} [\ln (\frac{{\dot{u}}^{2}}{V_{c r}^{2}} + 1)] \end{matrix}

(8)

\begin{matrix} ε_{d}^{L} = \int_{0}^{\dot{u}} f_{d}^{L} d \dot{x} = \int_{0}^{\dot{u}} c_{d, e q} \dot{x} d \dot{x} = \frac{1}{2} c_{d, e q} {\dot{u}}^{2} \end{matrix}

(9)

The error or difference between the dissipated energy of the nonlinear and linearized system is

\begin{matrix} e = ε_{d}^{N L} - ε_{d}^{L} = F_{d m a x} V_{c r} [\ln (\frac{{\dot{u}}^{2}}{V_{c r}^{2}} + 1)] - \frac{1}{2} c_{d, e q} {\dot{u}}^{2} \end{matrix}

(10)

By minimizing the mean square error, one has the energy-based linearization coefficient $c_{d, e q}^{E B}$

\begin{matrix} \frac{\partial}{\partial c_{d, e q}} E [e^{2}] = 0 \to c_{d, e q}^{E B} = 2 F_{d m a x} V_{c r} \frac{E [{\dot{u}}^{2} \ln (\frac{{\dot{u}}^{2}}{V_{c r}^{2}} + 1)]}{E [{\dot{u}}^{4}]} \end{matrix}

(11)

Assuming the probability density function of the velocity response is a zero-mean Gaussian, one has the following expressions

\begin{matrix} c_{d, e q}^{F B - G} = \frac{2 F_{d m a x} V_{c r}}{σ_{\dot{u}}^{2}} (\sqrt{π} β (e r f (β) - 1) e^{β^{2}} + 1) \end{matrix}

(12)

\begin{matrix} c_{d, e q}^{E B - G} = 2 F_{d m a x} V_{c r} \frac{E [{\dot{u}}^{2} \ln (\frac{{\dot{u}}^{2}}{V_{c r}^{2}} + 1)]}{E [{\dot{u}}^{4}]} \\ = \frac{2 F_{d m a x} V_{c r}}{3 σ_{\dot{u}}^{2}} (- 2 \sqrt{π} β e^{β^{2}} + π e r f i (β) - 2 \ln (2 β) - γ + 2) \\ + \frac{4 F_{d m a x} V_{c r}}{3 σ_{\dot{u}}^{2}} (h y p e r g e o m ([1,1], [\frac{1}{2}, 2], β^{2}) β^{2}) \end{matrix}

(13)

where

β = V_{c r} / (\sqrt{2} σ_{\dot{u}})

e r f (\cdot)

is the error function,

e r f i (\cdot)

is the imaginary error function,

h y p e r g e o m (\cdot)

is the hypergeometric function, and

γ

is the Euler-Mascheroni constant. Noting that

c_{d, e q}

depends on

σ_{\dot{u}}

, while

σ_{\dot{u}}

is an unknown which implicitly depends on

c_{d, e q}

, an iterative procedure in the frequency domain is necessary. By using an arbitrary value of

c_{d, e q}

, the variance of velocity can be evaluated as

\begin{matrix} σ_{\dot{u}}^{2} = \int_{0}^{\infty} ω^{2} {| H (ω) |}^{2} S_{{\ddot{u}}_{g}} (ω) d ω \end{matrix}

(14)

Here $H (ω)$ is the transfer function of the linearized equation (5)

\begin{matrix} H (ω) = {(ω_{0}^{2} - ω^{2} + 2 i ξ^{(e)} ω_{0} ω)}^{- 1} \end{matrix}

(15)

with i being the imaginary unit, and effective damping ratio

ξ^{(e)} = (c + c_{d, e q}) / 2 m ω_{0}

. An approximate expression for the velocity response variance is

\begin{matrix} σ_{\dot{u}}^{2} ≅ \frac{π S_{{\ddot{u}}_{g}} (ω_{0})}{4 ξ^{(e)} ω_{0}} \end{matrix}

(16)

Stochastic seismic analysis for MDOF structure

The extension of the above stochastic linearization technique in the ECD-SDOF system to the case of MDOF systems is considered. The building is modeled as a multi-story shear-type structure. Suppose that the damping ratio of the primary frame structure (without energy dissipation devices) is 5% with Rayleigh damping. The graphical representation of the damped structure is shown in Figure 2.

Figure 2.

Schematic of MDOF structure equipped with ECDs.

The equation of motion for the multi-degree of freedom structural system is²⁰

\begin{matrix} [M] {\ddot{u}} + [C^{s}] {\dot{u}} + [K] {u} + {F_{d}} = - [M] {I} {\ddot{u}}_{g} . \end{matrix}

(17)

Here $[M]$ , $[C_{s}]$ , and $[K]$ represent the mass, inherent structural damping, and stiffness matrices, respectively. ${\ddot{u}}$ , ${\dot{u}}$ , and ${u}$ are the acceleration, velocity, and displacement vectors at the degrees of freedom. ${I}$ is the unit vector. ${F_{d}}$ is the vector of additional force due to damper devices whose j th component $F_{d, j}$ is

\begin{matrix} F_{d, j} = 2 F_{d m a x, j} {(\frac{V_{c r, j}}{{\dot{u}}_{j} - {\dot{u}}_{j - 1}} + \frac{{\dot{u}}_{j} - {\dot{u}}_{j - 1}}{V_{c r, j}})}^{- 1} - 2 F_{d m a x, j + 1} {(\frac{V_{c r, j + 1}}{{\dot{u}}_{j + 1} - {\dot{u}}_{j}} + \frac{{\dot{u}}_{j + 1} - {\dot{u}}_{j}}{V_{c r, j + 1}})}^{- 1} . \end{matrix}

(18)

According to SLT, the equivalent linear system is assumed as

\begin{matrix} [M] {\ddot{u}} + [C^{e}] {\dot{u}} + [K] {u} = - [M] {I} {{\ddot{u}}_{g}} . \end{matrix}

(19)

Here the elements of equivalent linear damping matrix are

\begin{matrix} c_{j, j}^{e} = c_{j, j}^{s} + 2 F_{d m a x, j} V_{c r, j} \frac{E [{(1 + \frac{V_{c r, j}^{2}}{{({\dot{u}}_{j} - {\dot{u}}_{j - 1})}^{2}})}^{- 1}]}{E [{({\dot{u}}_{j} - {\dot{u}}_{j - 1})}^{2}]} + 2 F_{d m a x, j + 1} V_{c r, j + 1} \frac{E [{(1 + \frac{V_{c r, j + 1}^{2}}{{({\dot{u}}_{j + 1} - {\dot{u}}_{j})}^{2}})}^{- 1}]}{E [{({\dot{u}}_{j + 1} - {\dot{u}}_{j})}^{2}]} \end{matrix}

(20)

\begin{matrix} c_{j, j + 1}^{e} = c_{j + 1, j}^{e} = c_{j, j + 1}^{s} - 2 F_{d m a x, j + 1} V_{c r, j + 1} \frac{E [{(1 + \frac{V_{c r, j + 1}^{2}}{{({\dot{u}}_{j + 1} - {\dot{u}}_{j})}^{2}})}^{- 1}]}{E [{({\dot{u}}_{j + 1} - {\dot{u}}_{j})}^{2}]} . \end{matrix}

(21)

When only the kth mode of the undamped system $ϕ_{k}$ is activated, the corresponding structural response ${u_{k}}$ reads as

\begin{matrix} {u_{k}} = {ϕ_{k}} X_{k} \end{matrix}

(22)

By equations (20) and (21), the equivalent damping elements can be written as

\begin{matrix} c_{j, j}^{e} = c_{j, j}^{s} + 2 F_{d m a x, j} V_{c r, j} \frac{E [{(1 + \frac{V_{c r, j}^{2}}{Δ_{k, j}^{2} X_{k}^{2}})}^{- 1}]}{E [Δ_{k, j}^{2} X_{k}^{2}]} + 2 F_{d m a x, j + 1} V_{c r, j + 1} \frac{E [{(1 + \frac{V_{c r, j + 1}^{2}}{Δ_{k, j + 1}^{2} X_{k}^{2}})}^{- 1}]}{E [Δ_{k, j + 1}^{2} X_{k}^{2}]} \end{matrix}

(23)

\begin{matrix} c_{j, j + 1}^{e} = c_{j + 1, j}^{e} = c_{j, j + 1}^{s} - 2 F_{d m a x, j + 1} V_{c r, j + 1} \frac{E [{(1 + \frac{V_{c r, j + 1}^{2}}{Δ_{k, j + 1}^{2} X_{k}^{2}})}^{- 1}]}{E [Δ_{k, j + 1}^{2} X_{k}^{2}]} \end{matrix}

(24)

\begin{matrix} Δ_{k, j} = ϕ_{k, j} - ϕ_{k, j - 1} \end{matrix}

(25)

$ϕ_{k, j}$ is the j th component of vector $ϕ_{k}$ .

Let $X_{k}$ be the response of the k th oscillator in modal coordinates. Then, it holds

\begin{matrix} {\ddot{X}}_{k} + 2 ξ_{k}^{e} ω_{k} {\dot{X}}_{k} + ω_{k}^{2} X_{k} = - P_{k} {\ddot{u}}_{g} \end{matrix}

(26)

where

P_{k} = \frac{{ϕ_{k}}^{T} [M] {I}}{{ϕ_{k}}^{T} [M] {ϕ_{k}}}

is the kth modal participation factor. kth modal equivalent damping ratio

ξ_{k}^{e}

can be obtained from equations (23) and (24), which is

\begin{matrix} ξ_{k}^{e} = ξ^{s} + 2 \sum_{j = 1}^{n} (F_{d m a x, j} V_{c r, j} \frac{E [{(1 + \frac{V_{c r, j}^{2}}{Δ_{k, j}^{2} X_{k}^{2}})}^{- 1}]}{E [Δ_{k, j}^{2} X_{k}^{2}]} + F_{d m a x, j + 1} V_{c r, j + 1} \frac{E [{(1 + \frac{V_{c r, j + 1}^{2}}{Δ_{k, j + 1}^{2} X_{k}^{2}})}^{- 1}]}{E [Δ_{k, j + 1}^{2} X_{k}^{2}]}) ϕ_{k, j}^{2} \\ - 4 \sum_{j = 2}^{n} (F_{d m a x, j} V_{c r, j} \frac{E [{(1 + \frac{V_{c r, j}^{2}}{Δ_{k, j}^{2} X_{k}^{2}})}^{- 1}]}{E [Δ_{k, j}^{2} X_{k}^{2}]}) ϕ_{k, j} ϕ_{k, j - 1} \end{matrix}

(27)

From equation (16), the approximation of the k th modal response is

\begin{matrix} σ_{{\dot{X}}_{k}}^{2} ≅ \frac{π P_{k} S_{{\ddot{u}}_{g}} (ω_{k})}{4 ξ^{e} ω_{k}} \end{matrix}

(28)

Numerical analysis

The accuracies of the two different SLT equivalent criteria are evaluated by comparisons with the time domain response of eddy current damping SDOF system. The response of this nonlinear system can be obtained by Monte Carlo simulations (MCSs) on the Kanai-Tajimi function and numerical integration.

Assuming firm soil conditions, the PSD function’s filter parameters can be defined as $ω_{g} = 15 r a d / s$ , $ζ_{g} = 0.6$ , $ω_{f} = 1.5 r a d / s$ , and $ζ_{f} = 0.6$ and peak ground acceleration ${\ddot{u}}_{g 0}$ = 0.3g. To verify the correctness of the MCS, the earthquake acceleration from MCS is shown in Figure 3(a) and the comparison between Kanai-Tajimi function and PSD estimation of the MCS data is completed in Figure 3(b).

Figure 3.

PSD and simulated ground accelerations. (a) PSD and (b) stochastic ground accelerations by MCS.

A linear system with inherent damping ratio equal to 0.05 is utilized to test the correctness of calculations between frequency domain and time domain. In frequency domain calculations, equation (14) is the precise integral and equation (16) is the approximation. These two equations are adopted to compare the correctness and accuracy with time domain calculation results, which are shown in Figure 4. We can find that these two equations both have excellent accuracy while frequency domain integral does better.

Figure 4.

Standard deviation of velocity of linear system under different periods.

As for the parameter of eddy current dampers, the non-dimensional peak damping force $F_{d m a x} / m g$ =0.01, 0.03 and the critical velocity $V_{c r}$ =0.5m/s. To compare the accuracies of these two versions of SLT under a broader range of structural natural periods with inherent damping ratio $ξ^{(e)}$ =0.05, both the standard deviations of displacement and velocity are illustrated in Figure 5 and the equivalent damping ratio results are shown in Figure 6. The results show that the force-based SLT and the energy-based SLT both perform with good accuracy, while the force-based SLT does better. However, when the critical velocity is large and standard deviation of response is small the numerical integration will cause an error.

Figure 5.

Standard deviation of structural response under different periods.

Figure 6.

Linearized damping coefficient under different periods.

To examine the performance of these two versions of SLT in a broader range of nonlinear parameters of the ECD, the standard deviation of the response for T = 1s is reported in Figure 7 and Figure 8 by varying the non-dimensional peak damping force $F_{d m a x} / m g$ and the critical velocity $V_{c r}$ , respectively. The response of system reduces with increasing non-dimensional peak damping force evidently, while it does not change with increasing critical velocity monotonically. When the critical velocity equals to 0.2m/s, force-based criterion SLT and energy-based criterion SLT both have good performance. Energy-based criterion SLT behaves less accuracy than force-based criterion SLT does when the critical velocity of eddy current damper is lower and non-dimensional peak damping force equals 0.02. One possible reason is that energy-based criterion SLT is more sensitive to nonlinearity of system.

Figure 7.

Standard deviation of structural response for T = 1s under varying non-dimensional peak damping force $F_{d m a x} / m g$ .

Figure 8.

Standard deviation of structural response for T = 1s under varying critical velocity $V_{c r}$ .

As for the MDOF structure with ECDs, the force-based SLT is applied due to its better accuracy. All three floors of building model have the same mass and stiffness as 80 tons and 40000 kN/m, respectively. Both the peak damping force and the critical velocity are equal for all ECDs, where the critical velocity equals to 0.2m/s. In order to assess the formulas for MDOF system derived in Section 3, the Monte Carlo simulations based on the real nonlinear system are conducted. The standard deviation of the 1^st modal response for MDOF system is illustrated in Figure 9 within a range of varying the non-dimensional peak damping force $F_{d m a x} / m g$ from 0 to 0.05. Results show a good consistency between analysis conducted by the SLT and the MCS. The response from SLT is slightly smaller than that from MCS when peak damping force is relatively low, while it becomes greater than MCS result with a larger peak damping force.

Figure 9.

Standard deviation of structural response under varying non-dimensional peak damping force $F_{d m a x} / m g$ .

Conclusions

Structural response evaluation of linear system with ECD is intricate due to its nonlinear constitutive behavior. By utilizing the stochastic linearization technique, the relationship between the equivalent linearized damping coefficient and nonlinear eddy current damper’s parameters is derived under the stochastic ground motion characterized by a power spectral density function. The linearized structural response is validated to accurate within a broad range of parameters of eddy current dampers. Comparisons between the force-based criterion and the energy-based criterion are conducted. It reveals that force-based SLT has better accuracy than the energy-based one, especially with the relatively lower critical velocity of ECD causing system behaves more evident nonlinear property. The results show that SLT has a good performance for SDOF as well as MDOF system. In addition, the critical velocity of ECD has its optimal value under stationary stochastic earthquake with the fixed maximum damping force. It should be noted that the results presented in this paper are based on the assumption of the stationarity of both the seismic input and system response. There is a tendency that combining nonlinear eddy current damping technique with tuned mass or inerter type dampers, and the linearization research of nonlinear ECD is the basis.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Zhao-Yu Huo

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