PSI - Issue 64

Amir Shamsaddinlou et al. / Procedia Structural Integrity 64 (2024) 360–367 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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triangle. This method involves assigning colors to vertices and using colored dice to guide the fractal's development. In a similar vein, the Chaos Game Optimization (CGO) algorithm applies these principles to optimize solutions within a Sierpinski triangle's framework, mapping the search space to this triangle and evaluating potential solutions (seeds) to the Global Best (GB) , Mean Group ( ), and a selected seed ( ), thus combining fractal generation with optimization processes. These temporary triangles are instrumental in pinpointing viable new seeds within the Sierpinski framework. Each such triangle represents the i - th iteration and encompasses n potential seeds, determined via a combination of the dice's outcome and the three vertex points of the Sierpinski triangle: GB , and . The methodical details of this model are outlined as follows: 1 = + ×( × − × ), = 1,2, … , . (23) 2 = + ×( × − × ), = 1,2, … , . (24) 3 = + ×( × − × ), = 1,2, … , . (25) 4 = ( = + ), = 1,2, … , . (26) which, symbolizes a randomly chosen factor that addresses the bounds of seed movement. Further, and are random integers, either 0 or 1, reflecting the results one might get from rolling a dice. The mutation operator is another crucial aspect utilized to modify the position of eligible seeds as per equation (26). Also, k is a random integer falling within the range [1, d], whereas R represents a random number uniformly selected from the range [0, 1]. Within the CGO algorithm's mechanics, the search process's exploration and exploitation phases are adjusted using , which takes into account the limitations on the movement of seeds, enabling the following equation: = { 2 × ( ( × × ) +) +( ~1 ) (27) is a random number uniformly distributed in [0,1], while and are random integers ranging from 0 to 1. 6. Numerical Study This study examines the effectiveness of optimal TMD design strategies using a benchmark example, focusing on pole placement and metaheuristic methods. A ten-story shear building frame Kaveh et al. (2015) has been selected to evaluate the optimal design of TMD. Mechanical properties of stories represent this structural model, including mass, stiffness, and damping. The structural behavior of this model is assumed linear, where the elastic stiffness is = 650×10 3 kN/m . The mass of each story is 360 tons (1 ton = 9810 N), so based on the mass and elastic stiffness, the natural frequency is = 1.0108 . Furthermore, assuming a 5% damping ratio for the first vibration mode of the structural model, the linear viscous damping coefficient c has been considered 6200 kN.s/m. Table 1 shows the mechanical properties of the structural model for each story. Table 1. Mechanical properties of the ten-story building structures. Story Mass ( ×10 3 ) Stiffness ( ×10 3 ⁄ ) Damping ( ×10 3 . ⁄ ) 1-10 360 650 6.2 The position of the system poles for the uncontrolled structure is shown in Table 2. Indeed, the optimal state for building a structure equipped with TMD is achieved when the position of these poles is shifted to the left of the real axis of the s plane. The optimum design problem for TMD is conducted in an example with three different objective functions. In this example, the mechanical parameters of the TMD are assumed to be design variables. The lower and upper bounds values for the mass of the damper are between 0.02 (72 tons ) and 0.04 (144 tons ) of the total mass of the structure. The lower and upper bounds are selected as 0 and 5000 ⁄ for stiffness, respectively. Also, the lower and upper bounds values are defined as 0 and 1000 . ⁄ for damping, respectively. Also, the objective functions are considered as controlled drift response, the transfer function of acceleration response, and the pole placement scheme. In the design based on the seismic responses and evaluation of the seismic behavior, a white noise base excitation with a PGE of 0.4 g and a period of 40 seconds was used. Table 2. The location of poles in the s plane for the structural model.