Several passive energy dissipation devices have been implemented in practice as the seismic protective systems to mitigate structural damage caused by earthquakes. The solid viscoelastic dampers are among such passive energy dissipation systems. To examine the response reducing effectiveness of these dampers, it is necessary that engineers are able to conduct response analysis of structures installed with added dampers accurately and efficiently. The main objective of this work, therefore, is to develop formulations that can be effectively used with various models of the viscoelastic dampers to calculate the seismic response of a structure-damper system. To incorporate the mechanical effect from VE dampers in the structural dynamic design, it is important to use a proper force-deformation model to correctly describe the frequency dependence of the damper. The fractional derivative model and the general linear model are capable of capturing the frequency dependence of viscoelastic materials accurately. In our research, therefore, we have focused on the development of systematic procedures for calculating the seismic response for these models. For the fractional derivative model, we use the G1 and L1 algorithms to derive various numerical schemes for solving the fractional differential equations for earthquake motions described by acceleration time histories at discrete time points. For linear systems, we also develop a modal superposition method for this model of the damper. This superposition approach can be implemented to obtain the response time history for seismic input defined by the ground acceleration time history. For random ground motion that is described stochastically by the spectral density function, we derive an expression based on random vibration analysis to compute the mean square response of the system. It is noted that the numerical computations involved with the fractional derivative model can be complicated and cumbersome. To alleviate computation difficulty, we explore the use of a general linear model with Kelvin chain analog as a physical representation of the damper properties. The parameters in the model are determined through a curve fitting optimization process. To simplify the analytical work, a self-adjoint system of state equations are formulated by introducing auxiliary displacements for the internal elements in the Kelvin chain. This self-adjoint system can then be solved by using the modal superposition method, which can be extended to develop a response spectrum approach to calculate the seismic design response for the structural system for seismic inputs defined by design ground response spectra. Numerical studies are carried out to demonstrate the applicability of these formulations. Results show that all the proposed approaches provide accurate response values, and the response reduction effects of the viscoelastic dampers can be evaluated to assess their performance using these models and methods. However, the use of a general linear model of the damper is the most efficient. It can capture frequency dependence of the storage and loss moduli as well as the fractional derivative model. The calculation of the response by direct numerical integration of the equations of motion or through the use of the modal superposition approach is significantly simplified, and response spectrum formulation for the calculation of seismic response of design interest can be conveniently formulated. / Ph. D.

Yes / Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

This thesis aims to explore and compare the performance of fractional proportional-integrative-derivative controllers in the DC motor system and the fractional leaky integrate and fire (FLIF) model for neuron spike signals. The objective is to determine whether FPID controllers exhibit superior performance and significant improvements in control compared to PID controllers. In the study, FPID controllers were found to be crucial for effectively controlling complex systems with nonlinear or long memory behavior, addressing challenges that regular PID controllers struggle to handle. The results indicate that in the DC motor system, both FPID and PID controllers performe similarly, without significant differences in their effectiveness. However, when it comes to controlling and simulating neuron spike signals using the FLIF model, the FPID controller demonstrates significant improvements compared the PID controller. These findings suggest that while FPID controllers may not universally outperform PID controllers, they do offer advantages and improvements in certain scenarios. The effectiveness of FPID controllers is particularly highlighted in the context of the FLIF model, where they show significant enhancements in control compared to PID controllers.

No / In this work, we discuss two new multicarrier modulating kernels that can be adopted for multicarrier signaling. These multicarrier transforms are the fractional Forurier transform (FrFT) and discrete wavelet transforms (DWT). At first, we relate the transforms in terms of mathematical relationships, and then using numerical and simulation comparisons we show their performances in terms of bit error ratio (BER) for Multiple Input Multiple Output (MIMO) applications. Numerical results using BPSK and QPSK support that both can be applied for multicarrier signaling, however, it can be resource effective to drive the DWT as the baseband multicarrier kernel at the expense of the FrFT

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