Fractional order PID based optimal control for fractionally damped nonlocal nanobeam via genetic algorithm

Mathematical modeling of a fractionally damped nonlinear nanobeam via nonlocal continuum approach

“…However, this model with forcing function is having some inherent deficiencies that lead to erroneous results when considering the higher modes in predicting the dynamic behavior. These deficiencies are vividly pointed out in literature 34,54 where the authors have clearly shown that the Galerkin discretization method fails to predict the correct dynamics of the original system with forcing function described by partial differential equation along with boundary conditions. Rather, if the same problem is directly attacked by a reduction method namely, method of multiple scales (MMS) or method of averaging, can accurately predict the dynamic behavior of the nanobeam as no information related to nonlinearity is lost during the MMS.…”

“…The integro-partial differential equation (1) is difficult to solve generally. Making use of Green function of differential operator, the nonlocal differential form of constitutive relations considering the one-dimensional stress state in accordance with the Hooke’s law and nonlocal continuum theory may be expressed in the following form 16,34,35 where ∇ is the Laplacian operator, ∇2≡∂ 2∂x2 and E is the Young’s modulus of the beam material.…”

“…for free vibration study, the obtained mode shape for first mode using two methods are found to be perfectly matched and for the higher modes (second and third) the mode shapes are slightly shifted. Since the aim of the present work is to study the free vibration of nanobeam, so instead of focusing on these complex nonlinear phenomena, we only consider the free vibration where the factors mentioned in the list of references 34,54 may not pose any severe deficiency.…”

“…For more details, one may refer to the literature. 34,51 The Caputo-type fractional derivative adopted for the present study can be expressed as This term can be further replaced by a combination of linear damping and stiffness force as follows The following formulae are useful for the derivation of C(α) and K(α) as The displacement response of the nanobeam model given by equation (18) can be assumed to be Multiplying equation (37) with sinϕ and integrating with respect to ϕ with the limits 0 and 2π,one may obtain By making use of equation (36), equation (40) can be further expressed as Using a suitable transformation followed by a few mathematical manipulations, one may get In a similar way, one can also obtain the stiffness (K(α)) as Combining these two expressions (equations (42) and (43)), the fractional derivative term can be written as follows …”

Adaptive prescribed‐time disturbance observer using nonsingular terminal sliding mode control: Extended Kalman filter and particle swarm optimization

“…Consequently, many studies have been conducted to study the vibration behaviour of micro/nanobeams [31][32][33][34][35]. Furthermore, the analysis of vibration control is used to prevent the micro/ nanobeams from damage and improve the performance and resolution of beams [36][37][38][39]. However, the measured dimensions of nanobeams may not be exact; therefore, the estimation analysis and robust controllers are useful to enhance the accuracy.…”