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Simultaneous optimization of force and placement of friction dampers under seismic loading a
b
Letícia Fleck Fadel Miguel , Leandro Fleck Fadel Miguel & Rafael b
Holdorf Lopez a
Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil b
Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, Brazil Published online: 30 Mar 2015.
Click for updates To cite this article: Letícia Fleck Fadel Miguel, Leandro Fleck Fadel Miguel & Rafael Holdorf Lopez (2015): Simultaneous optimization of force and placement of friction dampers under seismic loading, Engineering Optimization, DOI: 10.1080/0305215X.2015.1025774 To link to this article: http://dx.doi.org/10.1080/0305215X.2015.1025774
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Engineering Optimization, 2015 http://dx.doi.org/10.1080/0305215X.2015.1025774
Simultaneous optimization of force and placement of friction dampers under seismic loading Letícia Fleck Fadel Miguela∗ , Leandro Fleck Fadel Miguelb and Rafael Holdorf Lopezb
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a Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil; b Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, Brazil
(Received 29 August 2014; accepted 27 February 2015) It is known that the use of passive energy-dissipation devices, such as friction dampers, reduces considerably the dynamic response of a structure subjected to earthquake ground motions. Nevertheless, the parameters of each damper and the best placement of these devices remain difficult to determine. Some articles on optimum design of tuned mass dampers and viscous dampers have been published; however, there is a lack of studies on optimization of friction dampers. The main contribution of this article is to propose a methodology to simultaneously optimize the location of friction dampers and their friction forces in structures subjected to seismic loading, to achieve a desired level of reduction in the response. For this purpose, the recently developed backtracking search optimization algorithm (BSA) is employed, which can deal with optimization problems involving mixed discrete and continuous variables. For illustration purposes, two different structures are presented. The first is a six-storey shear building and the second is a transmission line tower. In both cases, the forces and positions of friction dampers are the design variables, while the objective functions are to minimize the interstorey drift for the first case and to minimize the maximum displacement at the top of the tower for the second example. The results show that the proposed method was able to reduce the interstorey drift of the shear building by more than 65% and the maximum displacement at the top of the tower by approximately 55%, with only three friction dampers. The proposed methodology is quite general and it could be recommended as an effective tool for optimum design of friction dampers for structural response control. Thus, this article shows that friction dampers can be designed in a safe and economic way. Keywords: friction dampers optimization; dynamic problem optimization; mixed discrete and continuous variables; backtracking search optimization algorithm (BSA); passive vibration control; seismic load
1.
Introduction
For a long time, reduction of vibration amplitudes has been the subject of study by many researchers. For example, the use of vibration absorbers dates back to the 1900s when Frahm, in 1909, proposed a kind of tuned mass damper (TMD). The Frahm model was applied to a main spring-mass without damping that was attached to a small spring-mass without damping to reduce the displacement of the main mass subjected to harmonic load. More recently, there has been a rapid increase in the development and application of passive energy-dissipation devices, such as viscoelastic dampers, viscous fluid dampers, metallic yield dampers and friction dampers (Soong and Dargush 1997). A growing number of these dampers has been installed in structures *Corresponding author. Email:
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around the world. For example, in the McConnell Library of Concordia University in Montreal, Canada, 143 friction dampers were employed; in the Funade Pedestrian Bridge tower in Osaka, Japan, an air damper type TMD was developed for the structure; and 260 viscoelastic dampers were incorporated into the Columbia SeaFirst Building in Seattle, USA (Soong and Dargush 1997). The objective of these devices is to absorb a portion of the input energy, due to earthquake or wind, for instance, reducing the dynamic response of the structure. Despite the known effectiveness of these dampers in reducing the dynamic response, as been shown in many works (e.g. Filiatrault 1985; Qu, Chen, and Xu 2001; Rocha, Riera, and Miguel 2004; Miguel, Curadelli, and Riera 2004; Curadelli, Riera, and Miguel 2006; Miguel, Riera, and Curadelli 2006; Min, Seong, and Kim 2010), the development of methods for optimum use of these devices is still an important research issue. To enable utilization of these dampers in an economic way, several researchers, especially in the past decade, have started to study the optimization of their parameters and their best positions in a structure. Although a great number of articles on optimization of TMD exists (e.g. Chen and Wu 2001; Li and Qu 2006; Lee et al. 2006; Desu, Deb, and Dutta 2006; Warnitchai and Hoang 2006; Ghosh and Basu 2007; Hoang, Fujino, and Warnitchai 2008; Wang, Lin, and Lian 2009; Marano, Greco, and Chiaia 2010; Dehghan-Niri, Zahrai, and Mohtat 2010; Arfiadi and Hadi 2011; Farshi and Assadi 2011; Mohebbi et al. 2013; Fadel Miguel, Lopez, and Miguel 2013a; Lavan and Daniel 2013; Brzeski, Perlikowski, and Kapitaniak 2014), along with some works on the optimum design of viscous and viscoelastic dampers (e.g. Singh and Moreschi 2002; Movaffaghi and Friberg 2006; Aydin, Boduroglu, and Guney 2007; Marano, Trentadue, and Greco 2007; Aydin 2012; Sonmez, Aydin, and Karabork 2013), the authors have not found any article dealing with the problem of optimization of friction dampers in structures subjected to seismic action, in which both parameters, friction forces and placement (best location), are optimized simultaneously. However, some important investigations on the optimization of friction or hysteretic dampers, which deal with the optimization problem from different perspectives, contribute to the knowledge on the issue (e.g. Moreschi 2000; Jangid 2000; Uetani, Tsuji, and Takewaki 2003; Moreschi and Singh 2003; Basili and De Angelis 2007). Within this context, this article studies the optimization of a different type of passive control: friction dampers, for which there is a lack of studies in the literature. The main contribution of the present article is to propose a methodology to optimize at the same time the location of friction dampers and their friction forces in structures subjected to seismic loading, to achieve a desired level of reduction in the dynamic response. To the best of the authors’ knowledge, this is the first work to deal with this simultaneous optimization of friction dampers for passive control. Because the location of a friction damper at a particular position in a structure is a discrete number, it is a discrete design variable, whereas the friction forces of each damper are best represented by continuous numbers, i.e. they are continuous design variables. So, the optimization algorithm must assess a mixed-variable optimization problem that includes both discrete and continuous variables at the same time. Such problems are usually non-convex, and therefore must be solved by optimization methods capable of handling this type of problem. Heuristic algorithms are well suited to solving such optimization problems. The advantages of these algorithms include the following: (1) they do not require gradient information and can be applied to problems in which the gradient is difficult to obtain or simply does not exist; (2) they do not become stuck in local minima if correctly tuned; (3) they can be applied to non-smooth or discontinuous functions; (4) they furnish a set of optimal solutions instead of a single solution, giving the designer a set of options from which to choose; and (5) they can be easily employed to solve mixed-variable optimization problems (Miguel and Fadel Miguel 2012; Fadel Miguel, Lopez, and Miguel 2013b). Many heuristic algorithms have been presented in the recent literature, such as the bat algorithm (Yang and Gandomi 2012), cuckoo search (Gandomi, Yang, and Alavi 2013b) and
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krill herd (Gandomi, Alavi, and Talatahari 2013a; Gandomi et al. 2013), to name just a few. A review of nature-inspired algorithms can be found in Yang (2008, 2010). Among the heuristic algorithms, the backtracking search optimization algorithm (BSA), developed by Civicioglu (2013), has been shown to be very accurate and efficient (Civicioglu 2013); therefore, the BSA is selected for solving the optimization problem proposed in this article. Finally, it is important to note that uncertainties in the structural properties and/or in the seismic excitation can influence the optimum solution. Therefore, it would be ideal if these uncertainties were taken into account in the optimization procedure, leading to a problem of optimization under uncertainty, e.g. robust optimization (Greco, Lucchini, and Marano 2014; Lucchini et al. 2014) or reliability-based design (Taflanidis, Beck, and Angelides 2007). However, in this article a simplified model is adopted, which considers the problem to be deterministic in order to focus the attention on the proposed design methodology. From the practical point of view, the proposed approach may be extended to more complex models, e.g. robust optimization or reliability-based design of passive control devices. The main issue with this extension is the huge computational cost, which is already high for the deterministic problem. The proposed optimization procedure is followed by verification through numerical examples. This article is organized as follows: Section 2 presents the problem formulation, Section 3 describes the BSA, Section 4 presents the two illustrative examples, and Section 5 presents some conclusions.
2.
Problem formulation
This section presents the equation of motion, the simulation procedure of seismic loading and the proposed optimization problem. 2.1.
Equation of motion
The differential equation that governs the motion of a multi-degree-of-freedom system with added friction dampers and subjected to earthquake ground motions may be written as: f (t) + Kz(t) = −MBy¨ (t) Mz¨ (t) + Cz˙ (t) + F
(1)
in which M, C and K represent the n × n structural mass, inherent damping and stiffness matrices, respectively, and n is the number of degrees of freedom. The damping matrix C is considered to be proportional to the M and K matrices, as: C = aM + bK. z(t) is the n-dimensional relative displacement vector with respect to the base and a dot over a symbol indicates differentiation with respect to time. B is an n × d matrix of ground motion influence coefficients, i.e. this matrix contains the cosine directors of the angles formed between the base motion and the direction of the associated displacement with the considered degree of freedom. d is the number of considered ground motions (directions). y¨ (t) is a d-dimensional vector representing the seismic excitation, f (t) is the n-dimensional Coulomb friction force vector, which is i.e. the base acceleration. F defined by: f (t) = F fn sgn(v˙ (t)) F
(2)
where μ is the friction coefficient, assumed to be constant, and N is the fn = μN, in which F vector of normal force; v˙ (t) is the vector of relative velocity between the two ends of the friction
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damper and sgn(v˙ (t)) is the signal function, defined by:
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⎧ ⎪ ⎨+1 for v˙ (t) > 0 sgn(v˙ (t)) = 0 for v˙ (t) = 0 ⎪ ⎩ −1 for v˙ (t) < 0
(3)
As may be seen in Equations (2) and (3), the magnitude of the friction force is constant, but its direction is always opposite to that of the sliding velocity. In the problem under consideration in this article, i.e. determining and minimizing the dynamic response of structures with friction dampers, the direction of the sliding velocity changes often. These frequent changes in the velocity direction cause many discontinuities in the friction force, complicating the process of evaluating the response of systems with friction dampers. In this context, many authors have proposed alternative methods to represent the friction force. Tan and Rogers (1995) present an analysis of the friction energy dissipated by each mode to derive several models of equivalent friction modal damping. These friction damping factors can be subtracted from the overall modal damping factors when carrying out computer simulations where the dynamic friction forces are calculated directly. To verify whether the equivalent friction damping models are reasonable, Tan and Rogers (1995) simulated a two-degree-of-freedom system using three approaches: a piecewise continuous analytical method, numerical integration with a spring-damper friction model, and numerical integration with equivalent friction damping. Five cases with periodic and non-periodic excitation were considered. These authors concluded that the equivalent friction damping works very well for cases where sliding motions predominate, and for long periods of sticking, the overall motions are predicted well but the detailed motions are approximated. Mostaghel and Davis (1997) showed that the discontinuous Coulomb friction force can be represented by at least four different continuous functions. Each of these functions involves one constant (α i ) that controls the level of accuracy of that function’s representation of the friction force. The accuracy of the various representations was verified by these authors, comparing the response of a single-degree-of-freedom system, obtained through numerical solutions utilizing these representations, with an exact analytical solution. The four continuous functions analysed by Mostaghel and Davis (1997) are given in Equation (4) and represented in Figure 1. f2 (α2 , v˙ ) = Tanh(α2 v˙ ) f1 (α1 , v˙ ) = Erf(α1 v˙ ) f3 (α3 , v˙ ) = (2/π )ArcTan(α3 v˙ )
(4)
f4 (α4 , v˙ ) = α4 v˙ /1 + α4 |˙v| In this article, as well as in previous works (Miguel 2002; Miguel and Riera 2002; Curadelli, Miguel, and Riera 2003; Miguel and Riera 2008), the authors chose to use the nonlinear function f2 (α2 , v˙ ) = Tanh(α2 v˙ ), suggested by Mostaghel and Davis (1997). In this article it is assumed that α 2 = 1e10 for both examples. A computational routine was developed by the authors in MATLAB language for determining the dynamic response of structures with added friction dampers, i.e. for solving Equation (1). This developed program uses the finite difference explicit method, which is a direct method of integration of the motion equations in the time domain. In addition, as previously explained, the discontinuous signal function was replaced by the continuous hyperbolic tangent function, according to function f 2 of Equation (4).
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Figure 1. (1997).
2.2.
5
Comparisons of the four representations of the signal function (α i = 10) proposed by Mostaghel and Davis
Simulating seismic loading
To solve Equation (1), it is necessary to define the seismic loading. In the present article, as well as in many studies found in the literature (e.g. Singh and Moreschi 2001; Singh and Moreschi 2002; Mohebbi et al. 2013), the seismic load is defined as a one-dimensional earthquake loading that is simulated by passing a Gaussian white noise process through a Kanai–Tajimi filter (Kanai 1961; Tajimi 1960) with the power spectral density (PSD) function given by: S(ω) = S0
ωg4 + 4ωg2 ξg2 ω2 (ω2
−
ωg2 )2
+
4ωg2 ξg2 ω2
,
S0 =
0.03ξg π ωg (4ξg2 + 1)
(5)
in which S 0 is constant spectral density and ξ g and ωg are the ground damping and frequency, respectively. That is, this procedure enables the designer to construct the seismic records taking into account characteristics of the soil where the structure will be built. However, it is important to note that the optimal solution may vary if the spectral characteristics (ξ g and ωg ) of the Kanai–Tajimi spectrum are changed. The ground damping and frequency will depend on the characteristics of the region where the building is being or will be built, and should be carefully chosen by the designer. For illustrative purposes, it is considered that the building code of the region where the structures are to be built requires a peak ground acceleration (PGA) of 0.20 g. For the ground parameters, values of ξ g = 0.5 and ωg = 20 rad/s are adopted. The simulated time history of the Kanai–Tajimi excitation used for designing optimal friction dampers is shown in Figure 2. Figure 3 shows the PSD of this generated acceleration record. 2.3.
Placement and force optimization
Based primarily on an analogy to the automotive brake, Pall, Marsh, and Fazio (1980) began the development of passive frictional dampers to improve the seismic response of structures. The objective is to slow down the motion of buildings ‘by braking rather than breaking’ (Pall and Marsh 1982). Despite friction dampers having been used for around 30 years, a mathematical optimization procedure for their use, based on optimization algorithms, either does not exist or is
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Figure 2.
Kanai–Tajimi excitation with peak ground acceleration of 0.20 g (1.96 m/s2 ).
Figure 3.
Power spectral density of the generated Kanai–Tajimi excitation.
not widely known. Thus, in this section, the proposed method for the optimum design of friction dampers will be described. Common criteria used to assess the effectiveness of passive energy-dissipation devices, such as friction dampers, installed in structures are their ability to reduce the maximum displacement (zmax ) of the structure and to reduce the maximum interstorey drift (d max ) in the case of buildings. In this article, the optimization problem has as objective function to minimize the maximum interstorey drift for the case of the shear building (first example) and to minimize the maximum displacement for the transmission line tower (second example), determined through the calculation of the vector z(t), which is obtained solving Equation (1) in the time domain. The proposed method is flexible and the objective function can be changed easily, as long as it can be calculated numerically. The design variables are the friction forces of each friction damper (F fn ), considered as continuous variables, and the positions of each passive energy-dissipation device in the structure ), considered as discrete variables. The corresponding constraints are the allowed limits (vector P for the friction forces (lower bound ≤ F fn ≤ upper bound), the maximum number of dampers to be installed (nd ) in the structure, and the number and position of predefined possible locations for the dampers (np ). is expressed as an np -dimensional vector of position consisting of 0 and 1, Consequently, P which indicates that there is a damper in that position if the number is 1. Therefore, the maximum is nd . That is, presuming a maximum number of dampers, the number of ones in the vector P optimal variable shows different positions of number 0 and 1.
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Therefore, given all predetermined possible positions for the friction dampers in the structure and the desired maximum number of devices to be placed in the structure, it is of interest to determine the optimal location of each friction damper and the optimal friction forces of such devices to achieve a maximum reduction in the structural response, when the structure is excited by earthquake ground motions, using the BSA. For each optimization run, the program gives the optimal placement of the friction dampers and the optimal friction force of each damper, while constraining the maximum number of available dampers and predefined positions. For ]. Thus, the fn , P convenience of notation, the design variables are grouped into the vector x = [F optimization problem can be posed as: Find
x
J(x) = dmax (x) ou J(x) = zmax (x) ⎧ j max min ⎪ ⎨Ffn ≤ Ffn ≤ Ffn , j = 1, . . . , nd Subject to number of available positions = np ⎪ ⎩ maximum number of dampers = nd
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Minimizes
(6)
The optimization problem stated above can be solved using the BSA described in the next section.
3.
Backtracking search optimization algorithm
As explained previously, the optimization problem under consideration in the present article is a mixed-variable optimization problem. Thus, the optimization algorithm must be able to deal with a problem that includes both discrete and continuous variables at the same time. Such problems are usually non-convex, and therefore must be solved by optimization methods capable of handling this type of problem. Evolutionary algorithms are well suited to solve such optimization problems. Within this context, among the evolutionary algorithms, the BSA, developed recently by Civicioglu (2013), has been shown to be very accurate and efficient (Civicioglu 2013); therefore, the BSA is successfully used for solving the optimization problem proposed in this article. The bio-inspired philosophy of the BSA is analogous to the return of a social group of living creatures at random intervals to hunting areas that were previously found to be fruitful for obtaining nourishment. According to Civicioglu (2013), the development of the BSA was motivated by studies that attempt to develop simpler and more effective search algorithms. The BSA has many advantages over other search algorithms, such as: it has a single control parameter, namely mixrate; its performance in solving a problem is not oversensitive to the initial value of this control parameter; it has a simple structure that is effective, fast and capable of solving multimodal problems and that enables it to easily adapt to different numerical optimization problems; its strategy for generating a trial population includes two new crossover and mutation operators; its strategies for generating trial populations and controlling the amplitude of the search-direction matrix and search-space boundaries give it very powerful exploration and exploitation capabilities; it possesses a memory in which it stores a population from a randomly chosen previous generation for use in generating the search-direction matrix, and consequently, its memory allows it to take advantage of experiences gained from previous generations when it generates a trial preparation (Civicioglu 2013). In addition, Civicioglu (2013) compared 75 boundary-constrained benchmark problems and three constrained real-world benchmark problems, and showed that the BSA could solve the 78 benchmark problems more successfully than the comparison algorithms (six widely used evolutionary algorithms).
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The BSA is explained in detail in Civicioglu (2013), where in addition to a detailed description of the algorithm, the author presents the pseudo-codes and provides the path to download the algorithm for implementation (the software codes in MATLAB). Thus, in the present article only a brief description of the BSA is presented. For a comprehensive review of the algorithm the reader is referred to Civicioglu (2013). According to Civicioglu (2013), the BSA can be divided into five processes or steps: Initialization, Selection-I, Mutation, Crossover and Selection-II, which are described in the following subsections. 3.1.
Initialization
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The initial population PP of the BSA is given by: PPi,j ∼ U(lowj , upj )
(7)
for i = 1, 2, 3, . . . , N and j = 1, 2, 3, . . . , D, in which N is the population size and D is the problem dimension, i.e. the length of the variable vector x. U is the uniform distribution and each PPi is an individual in the population PP. 3.2.
Selection-I
The historical population (oldPP) to be used to calculate the search-direction matrix is determined in this step. The initial oldPP is obtained by: oldPPi,j ∼ U(lowj , upj )
(8)
The BSA has the option of redefining the oldPP at the beginning of each iteration through the following ‘if–then’ command: if
a
