Optimal design of multiple tuned mass dampers for controlling the earthquake response of randomly excited structures

Abstract

In this paper, the optimal design of multiple tuned mass dampers (MTMDs) is performed to mitigate the seismic response of structures when considering the earthquake excitation as a random process. Frequency-domain analysis of the structure is performed according to random vibration theory in order to establish a new optimization problem in terms of the parameters of tuned mass dampers (TMDs) as design variables. In this regard, a semi-analytical formulation is developed for the random earthquake response of structures equipped with tuned mass dampers, which is highly efficient for each objective function evaluation needed for optimal design. Hence, the present formulation renders an effective optimization problem for the rapid design of MTMDs under earthquake loading. Minimization of the maximum standard deviation of structural response is proposed as an objective function, while some constraints are defined for the parameters and response of TMDs. Five well-known metaheuristic methods including particle swarm optimization, whale optimization algorithm, grey wolf optimizer, water evaporation optimization, and bat algorithm are employed for solving the proposed optimization problem. A comprehensive design example is considered based on a ten-story building frame, which demonstrates the efficiency and capability of the proposed random vibration-based problem for optimal design of MTMDs. The optimized TMDs are also verified with response history analysis in the time domain using several seismic ground motions.

Appendix A

1.1 A.1 Bat algorithm

• Step 1 Initialization. First, the total number of bats is selected, and then the position of each bat is initialized randomly according to the search space. The position corresponds to a solution for the optimization problem.

• Step 2 Updating positions. Bats randomly move with a velocity of v_i from position x_i to a new one using a fixed frequency f, and loudness A₀ to find prey. For each bat, a position vector and a velocity vector are defined in the search space. A new solution (position, $x_{i}^{t}$) and a new velocity ($v_{i}^{t}$) of the ith bat in the iteration t are computed by:

  $$v_{i}^{t} = v_{i}^{t - 1} + f_{i} (x_{i}^{t - 1} - x_{{{\text{cgbest}}}} )$$

  (A1)
  $$x_{i}^{t} = x_{i}^{t - 1} + v_{i}^{t}$$

  (A2)
  $$f_{i} = f_{\min } + \beta (f_{\max } - f_{\min } )$$

  (A3)

  with $\beta$ being a uniform random number in [0,1], and $f_{i}$ denotes a frequency drawn from [f_min, f_max] wherein f_min and  f_max can be chosen as 0 and 2, respectively, depending on the problem. Also, $x_{{{\text{cgbest}}}}$ is the current global best position obtained after comparing all the solutions corresponding to all the bats.

• Step 3 Generating local positions. A solution among better solutions is selected randomly, and then a new local solution around the selected solution is generated by a random walk, as given by:

  $$x_{{{\text{new}}}} = x_{{{\text{old}}}} + \varepsilon \left\langle {A_{{}}^{t} } \right\rangle$$

  (A4)

  where $\varepsilon$ is a random number drawn from [− 1,1]; $\left\langle {A_{{}}^{t} } \right\rangle$ denotes the mean value of $A_{i}^{t}$ in the current iteration, respectively. However, these values can be adjusted by the considered optimization problem.

• Step 4 Accepting new positions. The new solution is acceptable when it is better than $x_{{{\text{cgbest}}}}$ under a condition on the loudness value. The loudness decreases (A_i) and the rate of pulse emission (r_i) increases once a bat finds its prey, as follows:

  $$A_{i}^{t + 1} = \alpha A_{i}^{t}$$

  (A5)
  $$\,r_{i}^{t + 1} = r_{i}^{0} [1 - \exp ( - \gamma t)]$$

  (A6)

  where $\alpha$ > 0 and $\gamma$ < 1 are constants. For example, A_min = 0 implies that a bat finds the prey and temporarily stops emitting any sound. The algorithm utilizes a dynamic strategy for exploration and exploitation. In fact, the alterations in pulse emission rates and loudness mainly control the exploration and exploitation abilities.

• Step 5 Stopping criterion. The bats in the current iteration are sorted according to the best solution, and then the best global solution is stored. Go to Step 2 until satisfying a stopping criterion.

1.2 A.2 Water evaporation optimization

• Step 1 Initialization. The parameters of the algorithm including the number of water molecules (nWM) and maximum number of iterations (maxNITs) are set.

• Step 2 Generating the water evaporation matrix. Every water molecule employs the rules of evaporation probability considered for each phase of the algorithm, as mentioned in Refs. [52, 54].

• Step 3 Generating the random permutation-based matrix of step size. The permutation-based matrix of step size is randomly formed by:

  $$stepsize = rand.(WM[permute1(i)(j)] - WM[permute2(i)(j)]$$

  (A7)

  in which rand shows a random number drawn from [0, 1]; permute1 and permute2 represent the permutation functions; also, i and j are the number of water molecules and the number of dimensions, respectively.

• Step 4 Generating evaporated water molecules and updating the matrix of water molecules. The evaporated set of water molecules newWM is formed by the current set of molecules WM plus the product of the step size matrix and evaporation probability matrix, as follows:

  $$newWM = WM + stepsize \times \left\{ {\begin{array}{*{20}c} MEP & \quad {NITs \le \ maxNITs/2} \\ DEP &\quad {NITs > \ maxNITs/2} \\ \end{array} } \right.$$

  (A8)

  where MEP and DEP are probability matrices of the monolayer evaporation and the droplet evaporation, respectively. The new water molecules are assessed by the corresponding objective function values. The old molecule i is replaced by a new molecule when it is better than the old one. The best water molecule is denoted by bestWM.

• Step 5 Stopping criterion. Once the total number of iterations of the algorithm (NITs) is greater than the maximum number of iterations (maxNITs), the process is terminated. Otherwise, the process is continued from Step 2.

1.3 A.3 Particle swarm optimization

• Step 1 Initialization. The parameters of the PSO algorithm are adjusted. Also, the initial positions of all particles are generated randomly and the initial velocities are chosen as zero. The objective function is evaluated for each particle to obtain pbest_i and gbest corresponding to the best-experienced position of the ith particle and the global best position in the swarm during the iterations performed.

• Step 2 Updating procedure. Velocities and positions of each particle are obtained by

  $$v_{i}^{k + 1} = wv_{i}^{k} + c_{1} r_{1} (pbest_{i}^{k} - x_{i}^{k} ) + c_{2} r_{2} (gbest_{{}}^{k} - x_{i}^{k} )$$

  (A9)
  $$x_{i}^{k + 1} = x_{i}^{k} + v_{i}^{k + 1}$$

  (A10)

  where $x_{i}^{k}$ and $v_{i}^{k}$ represent the position and the velocity of the ith particle for the kth iteration, respectively; $r_{1}$ and $r_{2}$ stand for two uniformly distributed random numbers in range $[0,1]$. Also, w shows a linear function known as the inertia weight decreased by increasing the iteration number. Of course, the inertia weight was introduced in the modified variant of the PSO to reduce the effects of previous velocities.

• Step 3 Stopping criterion. According to a stopping condition, the process can be terminated or can be repeated from Step 2.

1.4 A.4 Grey wolf optimizer

• Step 1 Initialization. Positions of agents are randomly obtained and their objective function values are calculated. The three best agents are assumed to be alpha, beta and delta in an order of their solution quality.

• Step 2 Updating process. Positions of agents are computed by the following relations:

  $$\begin{gathered} D_{\alpha } = \left| {C_{1} .x_{\alpha } - x} \right|, \hfill \\ D_{\beta } = \left| {C_{2} .x_{\beta } - x} \right|, \hfill \\ D_{\delta } = \left| {C_{3} .x_{\delta } - x} \right|, \hfill \\ \end{gathered}$$

  (A11)
  $$\begin{gathered} x_{1} = x_{\alpha }^{i} - A_{1} .D_{\alpha } , \hfill \\ x_{2} = x_{\beta }^{i} - A_{2} .D_{\beta } , \hfill \\ x_{3} = x_{\delta }^{i} - A_{3} .D_{\delta } , \hfill \\ \end{gathered}$$

  (A12)

  and

  $$x^{i} = \frac{{x_{1} + x_{2} + x_{3} }}{3}$$

  (A13)

  where a is a descending factor varying from 2 to 0 by increasing the iterations. A is a vector containing randomly generated numbers such that the wolves are compelled to attack the prey when the random numbers are greater than unity. Otherwise, an agent is enforced to move away from the prey. Vector $C$ includes random numbers providing the weights of prey to emphasize (if the number is larger than 1) or deemphasize (if the number is smaller than 1) the effect of prey. These parameters are due to exploration and exploitation abilities of the GWO. After the updating process, alpha, beta, and delta are redefined according to the solution quality calculated with objective function evaluations.

• Step 3 Stopping criterion. The process is finished when the GWO algorithm meets the convergence criterion. Otherwise, go to Step 2.

1.5 A.5 Whale optimization algorithm

• Step 1 Encircling prey. In the WOA, the best solution obtained until the current iteration is regarded as the target prey, while the other search agents try to update their positions toward that best agent. This kind of behavior is modeled by the following relations:

  $$\begin{aligned} & D_{{}} = \left| {C.x^{i,*} - x^{i} } \right| \\ & x^{i + 1} = x^{i,\,*} - AD \\ \end{aligned}$$

  (A14)

  where $x^{i,*}$ and $x^{i}$ denote the general best position and the current position of the whale, respectively, in the ith iteration; A and D are two random vectors.

• Step 2 Bubble-net attacking. In order to represent the bubble-net behavior of humpback whale, the positions of whales and prey are related using a spiral mathematical expression to imitate the helix-shaped movement of humpback whales, that is

  $$x^{i + 1} = D^{\prime} \cdot e^{bl} \cdot \cos (2\pi l) + x^{i,\,*}$$

  (A15)

  where

  $$D^{\prime} = \left| {x^{i,\,*} - x^{i} } \right|$$

  (A16)

  with b being a constant defining the shape of a logarithmic spiral, and l denotes a random number within [− 1,1].

• Step 3 Search for prey. To have a global exploration, when A is greater than 1 or less than − 1, the position of the agent is updated using a randomly selected agent (instead of the best agent):

  $$\begin{aligned} & D_{{}} = \left| {C.x_{rand}^{i} - x^{i} } \right| \\ & x^{i + 1} = x_{rand}^{i} - AD \\ \end{aligned}$$

  (A17)

  where $x_{rand}^{i}$ is a randomly selected whale (agent) in the current iteration (i). Further details of this algorithm have been provided in Ref. [39].