PSI - Issue 64

Amir Shamsaddinlou et al. / Procedia Structural Integrity 64 (2024) 360–367 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 where is the real part, and is the imaginary part. The roots and poles of the system are displayed in a coordinate system s with a horizontal axis of and a vertical axis . The open-loop pole – zero configurations and their root loci can be shown in s plane. A stable system must have all its closed-loop poles in the left-half the s plane. For a linear system, the stability or instability is a property of the system itself and does not depend on the system's input. The stability of a linear closed-loop system can be determined from the location of the closed-loop poles in the s plane. If all system poles are located on the left of the axis, any transient response is damped. So, the problem of absolute stability is solved by locating all poles in the left-half s plane. However, the complex-conjugate poles close to the axis cause the transient response with excessive oscillations. Therefore, to achieve well-damped transient response characteristics, the system's poles must lie in a specified distance of axe in the complex plane, such as the shaded region in Figure 1. Consequently, to design a stable system with a low transient response, the system's parameters are adjusted to obtain suitable pole configurations. In a tuned mass damper optimization problem, in which the mechanical properties of the passive control system are changed to reduce the structure's response, the desired structural system is assumed to be equivalent to a closed-loop system. Thus, the mechanical properties of TMD are determined in a way that the system's poles are located at the furthest distance from the axe. 4. Optimization Problem of TMD Based on Pole Placement The mechanical properties of TMD components, such as mass, stiffness, and damping, influence the performance of a structure equipped with a TMD. Therefore, the TMD design can be defined in the framework of an optimization problem. In this regard, meta-heuristic methods are generally used to optimize the design of TMD. These population based approaches solve the optimization problem by sampling in the search space. In these methods, unknown parameters are selected as design variables, and an objective function is minimized. Also, the problem's constraints are defined for design variables that should be within the acceptable domain. The formulation of the TMD design optimization problem is defined as: Find: Design Variables= { . } To minimize: { 1 = 1 = ∑ = 1 2 = 2 = . . 3 = 3 = ∑ = 1 Subject to: { ≤ ≤ ≤ ≤ ≤ ≤ (22) The objective function is based on the relatively controlled drift response, the transfer function (TF) of the acceleration response, and the pole placement scheme. In this numerical study, the CGO algorithm solves the optimization problem. Also, after the optimal design of the TMD using the proposed method, the seismic behavior of structures equipped with TMDs designed by different methodologies is compared. 5. Chaos Game Optimization (CGO) Fractal shapes, emerging from chaos systems, display self-similarity and repeating patterns at different scales, as Talatahari and Azizi (2021) illustrated through the chaos game technique, which produces fractals like the Sierpinski 363 4 Fig. 1. Suitable region for the placement of the poles.