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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 643927, 7 pages
Research Article

Reliability Analysis of Semiactive Magnetorheological Dampers Subjected to Harmonic Excitations

1Structural Department, Faculty of Civil Engineering, University of Alberta, Edmonton, AB, Canada
2Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran
3Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran

Received 9 April 2013; Accepted 21 May 2013

Academic Editor: Shengyong Chen

Copyright © 2013 N. Mohajer Rahbari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It has been seen that semiactive magnetorheological (MR) dampers are not sometimes reliable when employed in the building to contract acceleration demand of the stories. In the current paper, a three-storey sample building with semiactive MR dampers subjected to harmonic base excitation is taken under examination to clarify the reliability of such semiactive control systems in mitigation of absolute acceleration response of the buildings. Comparison between semiactively controlled building and uncontrolled original building is chosen as an implicit reliability limit-state function. Firstly, first-order reliability method (FORM) is engaged to examine the reliability of the system by linearizing the chosen nonlinear limit-state function. Afterwards, Monte Carlo simulation (MCS) is used to verify the obtained results by FORM method.

1. Introduction

Magnetorheological (MR) damper is a kind of semiactive control devices in which MR liquid is composed of magnetized tiny particles that are scattered in a mineral liquid such as silicon oil. While this liquid is exposed to a magnetic field, particles inside the liquid are polarized and chain is formed just in a few milliseconds. Hence, liquid will be changed into semi-solid state and will behave as a viscoplastic material which leads to alteration of yielding force (produced damping force).

The MR dampers are normally installed in a building so that they are resisting stories’ drift. To achieve an optimal performance of the system in terms of structural control, applied voltage of the current driver must be changed to conduct the damping force close to the calculated optimal control force at any moment according to the captured response feedback being analyzed by a predefined control algorithm.

Despite promising feature of using MR dampers, it has been seen that the semiactive magnetorheological (MR) dampers are not always reliable in mitigating the acceleration demand of the stories [1]. Hence, the application of the semiactive devices to the real civil structures still remains uncommon and unreliable to the engineers. In other words, there is an open question of why the building owners should go to the expense of an additional cost for application of semiactive MR devices as they counterproductively increase the acceleration response which may lead to a massive destruction of nonstructural components.

The interesting objective of the current work is to analytically assess the reliability of semiactive MR damper control systems in reduction capability of acceleration demand for buildings subjected to base acceleration by means of existing reliability methods. For this purpose, a three-storey sample shear building is taken under examination. Modified version of Bouc-Wen model [2] is used to predict the damper’s nonlinear restoring force, and clippedoptimal semiactive control algorithm [1] based on linear quadratic regulator (LQR) is updated to drive the MR damper during the control time. In terms of reliability analysis, the difference between acceleration response of controlled building and uncontrolled building resulting from time-history dynamic analysis is considered as the implicit limit-state function. This implies the failure of the control system, while it cannot reduce the acceleration demand compared with uncontrolled structure. Amplitude and frequency of harmonic excitation as well as stories’ stiffness are considered as random variables in reliability analysis to account for uncertainties that potentially exist in base excitation and structural natural characteristics, respectively. Well-established first-order reliability method (FORM) is applied to examine the reliability of the system by linearizing the nonlinear implicit limit-state function. Afterwards, Monte Carlo sampling (MCS) method is used to verify the obtained results by FORM analysis. All structural simulations and reliability analyses are coded in MATLAB platform.

2. Structural Model Simulation

The following sections describe the structural model and methods used to simulate the MR damper building casestudy under examination. The structural analytical model is coded in MATLAB platform.

2.1. Case-Study Introduction

An undamped three-storey shear building equipped with MR dampers is examined in this paper to evaluate the reliability of the system in acceleration response reduction. Structural properties of this building are given in Table 1.

Table 1: Structural properties of the case-study.
2.2. Excitation Introduction

Because a seismic ground acceleration can be expanded by infinite numbers of harmonic excitations, in the current study, we investigate the reliability of the system in response to a harmonic base acceleration with random properties as follows: in whichanddenote the base acceleration amplitude and frequency, respectively.

2.3. MR Damper Force

Herein, well-established Bouc-Wen phenomenological model (Figure 1) proposed by Spencer et al. [2] is utilized to predict the high nonlinear hysteretic damping force of the MR damper in the dynamic analytical model. In this case, according to Figure 1, MR damper’s force is calculated from the following equation: whereis a Bouc-Wen model shape parameter related to the MR material yield stress andis hysteretic displacement of model as given by in which , , , and are the hysteretic shape parameters. According to Figure 1, is defined by the following equation:

Figure 1: Mechanical model for MR damper [2].

To determine a comprehensive model that is valid for fluctuating magnetic fields, the coefficient, stiffness and viscous damping coefficientsand are defined as a linear function of the efficient voltagewith constant coefficients as given in the following equations:

To accommodate the dynamics involved in the MR fluid reaching rheological equilibrium, the following first-order filter is employed to calculate efficient voltage: where is the applied voltage of the current driver.

Here, one MR damper with the capacity of 1000 kN [3] is installed in such a way that causes a resistance to storey drift of the first storey. The model characteristics of the damper are reported in Table 2. Maximum input voltage () for this damper is equal to 10 V.

Table 2: Model parameters for 1000 kN MR damper [3].
2.4. System Description

The governing equation of motion of the controlled system can be expressed in state space as in whichis the vector of dampers force and denotes the state vector of the system that for an -storey building is written as and,, and are the constant matrices as follows: where demonstrates the number of stories with MR dampers, is base excitation influence vector (with unit components), and is dampers influence matrix with dimension.

2.5. Active Controller

To achieve optimal active control force at each time instant, linear quadratic regulator (LQR) algorithm with full state feedback is designed to minimize the following cost function: where and are dubbed as weighting matrices whose magnitudes are defined based on the relative significance of either structural response or control forces in the optimization procedure. And is the converter matrix.

Because the absolute acceleration of the top stories always dominant, herein, a cost function is chosen that weights the absolute acceleration of the top floor. Thus, matrixis obtained through the following equation:

Numerous controllers have been tested for the three-storey example under examination, and the best result was obtained by defining the weight matricesandas follows:

Now, the vector of optimal control force is calculated by solving the Riccati equation [4] as in whichis the Riccati matrix and is the feedback state of the system

2.6. Semiactive Clipped-Optimal Controller

After computation of the optimal control force () and calculation of the dampers force () according to feedback data, to approach the MR damper control force to the desired optimal force, at each time instant, applied voltage is set as the following description.(i)If these two forces are equal (), then applied voltage is not changed.(ii)If the absolute of MR damper force is less than the absolute of calculated optimal control force and both of them have the same sign, applied voltage should be increased to its maximum value.(iii)Otherwise the input voltage is set to zero.

Clipped-optimal algorithm can be summarized in the following equation: in whichdemonstrates the maximum applied voltage that is associated with saturation of magnetic field in MR damper andis the Heaviside function.

3. Random Variables

Two sets of general uncertainties including mechanical properties of the building and base excitation properties could be mainly considered for the current problem. In terms of structural uncertainties, the major contributor is the natural frequency because the building is modeled as an idealized lumped-mass multi-degree-of-freedom (MDOF) structure. Randomness of natural frequency could result from randomness in members geometry, modulus of elasticity, or mass. However, because the contribution of higher modes could be significant due to the variation of exciting frequency, instead of taking natural frequency as the random variable, the inter-storey stiffness of stories () considered as random variable to account for variation of natural frequency, and mass properties are held constant. Variability in structural dimensions and overall geometry tends to be small and can be adequately modeled by the normal distribution [5, 6]. However, stiffness cannot be modeled by normal distribution because it contains negative realizations. Hence, stiffness random variables are chosen to be modeled as lognormal distribution with mean of original building stiffness (Table 1) and coefficient of variation (C.O.V) of 5% because the objective of the current work is to assess the reliability of the semiactive MR damper control system for the particular problem under examination. Figure 2 indicates the lognormal probability density function (PDF) for the stiffness random variables.

Figure 2: Lognormal distribution of stories’ stiffness.

Moreover, the uncertainties for harmonic excitation come from its amplitude () and frequency (). Since the ground peak acceleration (GPA) of an individual site is normally determined by seismic risk analysis containing lots of assumptions and uncertainties, the amplitude of the harmonic excitation is taken as a more or less wide normally distributed random variable with mean value and covariance of 0.4 g and 10%, respectively. As base excitation contains loads of harmonic contributors with a variety of frequencies, the excitation frequency is modeled as a uniform random variable to take all significant harmonic terms into consideration with the same probability of occurrence. However, because the dynamic response of the building is under examination, frequency random variables rangeing from 0.1 to 3 times first natural frequency of the building to include all major possible dynamic loads (). Figure 3 illustrates the PDF for excitation amplitude and frequency. It should be verified that all five random variables are uncorrelated.

Figure 3: PDF for base excitation random variables.

4. Limit-State Function

To study the reliability of the aforementioned semiactive system in mitigation of the absolute acceleration response of the top storey which has been chosen as the control strategy, the difference between maximum absolute acceleration of the top storey in controlled and uncontrolled modes is considered as the limit-state function. In other word, failure mode is introduced when the semiactive MR damper control system fails to reduce the acceleration demand of the top storey in comparison with the original building. Hence, the limit-state function can be expressed as in which and denote the maximum acceleration of the top storey for the uncontrolled and controlled builds respectively, which will be obtained through time-history analysis of both ofthe operation modes. represents the vector of input random variables in each analysis which is defined as

5. Transformation of Random Variables and Jacobian Matrix

In reliability analysis, to obtain a workable space, it is usually required to repeatedly transform the realization of random variables from original distribution space into their corresponding standard normal space and vice versa. In the current problem, the random variables are independent of each other and uncorrelated. Thus, the following transformation equation is used to transform the vector of random variables to its corresponding standard normal random variables [7]: where is the cumulative distribution function (CDF) of the standard normal random variables and is the original CDF of the th random variable.

It is also necessary to obtain the transformation Jacobian to be used in transformation of gradient function for implicit limit-state function in the reliability analysis. The Jacobian matrix for uncorrelated random variables becomes a diagonal matrix obtained by [7] where is the PDF of the standard normal random variables and is the original PDF of the th random variable.

6. First-Order Reliability Method (FORM)

FORM reliability analysis method is based on the linearization of the limitstate at the design point which is the nearest point on the limit-state surface () to the origin in the standard normal space. This method is suitable for implicit limit-state functions and mostly results in conservative assessment. The FORM analysis method integrated with improved HLRF (iHLRF) algorithm [7] for finding new realization at each step of is implemented in MATLAB platform conforming to the following steps [8].(1) Opting a start set of random variables from the realization of their corresponding standard normal space ().Note that for the first step, the start point can be chosen from original distribution of random variables and then transformed into standard normal space.(2) Transformation of into the original random variables’ distribution space and evaluation of limit-state function .(3) Evaluation of the gradient () of the limit-state function through the following procedure: where for the current implicit function which is a function of maximum acceleration responses it comes to in which is the response vector defined as and gradient of response with respect to the random variables () is numerically obtained by taking finite differences of at each step of analysis, where denotes the standard deviation of random variable .(4) Examination of the following convergence criteria to verify that the found design point is being located on the limit-state surface while it possesses the closest distance from origin: in which (5) If convergence is satisfied, then probability index () and probability of failure () are obtained as If convergence is not met, the new random variables in standard normal space are found as in whichis the step size chosen as unity and is the search direction defined by and procedure is iteratively repeated from step 2 until the convergence criteria are satisfied.

7. Monte Carlo Simulation (MCS)

MCS reliability approach is based on repeated sampling from the linearization of the random variables and examining whether the limit-state function is located in failure region or not. Then, probability of failure is calculated based on taking these samples as independent random variables with equal probability of occurrence. The MCS method is coded in MATLAB platform following the following steps [8].(1)A set of random standard normal variables () is generated according to joint standard normal PDF as in which denotes the number of random variables.(2)Generated random standard normal numbers () are transformed to their original distribution space ().(3)Limit-state function is evaluated. And indicator function is defined as (4)The convergence criterion for covariance of failure probability () is checked as follows: in which is the total samples number and andrepresent standard deviation and mean values for indicator vector , respectively.(5)If the convergence criterion is achieved, the probability of failure is calculated as If not, more samples are required and the procedure is iteratively followed from step 1.

8. Numerical Results

8.1. FORM

Table 3 demonstrates the results obtained from FORM analysis. Due to inherent nonlinearity existing in limit-state function, various start points including mean values and mean values () plus or minus 1 to 3 standard deviations () were engaged to verify the robustness of the design point over the whole domain. As it can be seen, the FORM analysis converges to local design points for start points and . However, the rest of the 7 start points converge to nearly the same point which is the global design point for the introduced limit-state function. Hence, the dominant probability of failure obtained by FORM analysis comes to 16.10%. Table 4 depicts the design point values for random variables in their original space. It is apparently seen that the stiffness of the stories seems to possess less importance in reliability analysis, but the excitation frequency has the most importance.

Table 3: Results of FORM analysis.
Table 4: Random variables’ values in their original space corresponding to design point.
8.2. MCS

Figure 4 illustrates the covariance of failure probability () against number of iterations, and Figure 5 shows the changes of failure probability against iterations resulting from Monte Carlo simulation. The analysis converged at 17475th iteration with 1.9996% coefficient of variation for failure probability, and the probability of failure obtained equals 12.52%.

Figure 4: Covariance growth of failure probability.
Figure 5: Failure probability changes over iterations.

9. Conclusions

Reliability of semiactive MR damper control system in mitigation of absolute acceleration of a three-storey building was studied in the current work. From the analyses conducted by FORM and MCS methods, it can be concluded that MR dampers are almost reliable by more than 83.9% in terms of acceleration reduction of building due to base acceleration. The result achieved by MCS agrees with the correctness of the outcome of FORM analysis. MSC gives more accurate result by taking far more computation efforts compared with FORM analysis. However, FORM analysis develops a good conservative result just after finite number of iterations which means that FORM analysis could be effectively employed for this kind of problems.

Similar limit-state functions to (15) can also be defined for the other structural responses (e.g., interstorey drift and displacement responses) to obtain a more comprehensive reliability assessment. In this case, obtained results from FORM or MCS analyses for each limit-state function are considered as a “component reliability analysis.” Afterwards, component reliability analyses are integrated by means of “system reliability analysis” methods to acquire the whole system reliability [7]. Hence, system reliability analysis of the MR damper building is suggested as a future work.


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