We want to study the dynamics of a simple linear harmonic micro spring which is under the influence of the quantum Casimir force/pressure and thus behaves as a (an) nonlinear (anharmonic) Casimir oscillator. Generally, the equation of motion of this nonlinear micromechanical Casimir oscillator has no exact solvable (analytical) solution and the turning point(s) of the system has (have) no fixed position(s); however, for particular values of the stiffness of the micro spring and at appropriately well-chosen distance scales and conditions, there is (are) approximately sinusoidal solution(s) for the problem (the variable turning points are collected in a very small interval of positions). This, as a simple and elementary plan, may be useful in controlling the Casimir stiction problem in micromechanical devices.

Effective control strategies are necessary for enhancing wind turbine efficiency and lifespan. The purpose of this paper is to control the rotational speed of a wind turbine by adjusting the pitch angle of the blades. Initially, a one-degree-of-freedom closed-loop controller is designed based on the complementary sensitivity function, followed by a PID and a modified PID controller. Finally, their performance in tracking the desired rotational speed of the wind turbine, which includes step and sinusoidal signals, is compared. The 1DOF controller introduced oscillation, while the PID controller eliminated oscillation but exhibited steady-state error. To preserve the advantages of the PID controller and eliminate steady-state error, an integrator term (1/s) has been added to the controller. Consequently, the modified PID controller has been designed. The results of the modified PID controller show no steady-state error or peak overshoot percentage for step inputs. While, the PID and the one-degree-of-freedom closed-loop controller in tracking the step reference input result in steady-state errors of 1.464% and 0.497% and peak overshoot percentages of 23.688% and 84.389%, respectively. During the implementation of the one-degree-of-freedom closed-loop controller, significant oscillations occur while tracking the desired input. The oscillations occurring in the rotor speed can lead to increased fatigue in various wind turbine components, which is an undesirable outcome. However, no oscillations are observed in rotor speed with the PID and modified PID controllers. Finally, according to the obtained results, the modified PID controller demonstrates superior performance compared to the other controllers in tracking the desired rotor speed

In impulsive orbital maneuvers, a large disturbance torque is generated by the thrust vector misalignment from the center of mass (C.M). The purpose of this paper is to reject the mentioned disturbance and stabilize the spacecraft attitude, based on the combination of a one degree of freedom (1DoF) gimbaled-thruster, two reaction wheels (RWs) and spin-stabilization. In this paper, the disturbances are assumed to be unknown and reaction control systems (RCS) are not employed. The nonlinear two-body dynamics of the proposed system is formulated and validated by the Simmechanics model. The closed-loop controller includes a full state feedback controller based on the gimbal actuator, a self-tuning controller (STC) based on the two RWs and a least squares based disturbance estimator. The simulation results are given by which the applicability of the proposed method is illustrated.

In this paper, the efficient multi-step differential transform method (EMsDTM) is applied to get the accurate approximate solutions for strongly nonlinear duffing oscillator. The main improvement of EMsDTM which is to reduce the number of arithmetic operations, is thoroughly investigated and compared with the classic multi-step differential transform method (MsDTM). To illustrate the applicability and accuracy of the new method, six case studies of the free undamped and forced damped conditions are considered. The periodic response curves of both MsDTM and EMsDTM methods are obtained and contrasted with the exact solution or the numerical solution of Runge Kutta 4th order (RK4) method. This approach can be easily extended to other nonlinear systems and therefore is widely applicable in engineering and other sciences.

In this study, we present a semi-analytical technique known as the Optimal and Modified Homotopy Perturbation Method (OM-HPM) for solving nonlinear oscillators with time-dependent mass. The work extends existing approaches, including the standard Homotopy Perturbation Method (HPM), by introducing an auxiliary linear operator that minimizes residual error and enhances the method’s efficiency for both singular and non-singular nonlinear ordinary differential equations. The model of a harmonic oscillator with exponentially decaying mass is investigated using this method, and its equation of motion is derived using the Lagrangian formulation. The OM-HPM technique is applied to solve the resulting second-order nonlinear differential equation, and solutions are presented in series form. The method significantly reduces computational cost through the use of Newton-Cotes quadrature. Analytical illustrations demonstrate that the effectiveness of OM-HPM in solving complex nonlinear oscillatory systems.

In this paper, we consider the nonlinear equations with the additive white noise, which are commonly impossible to be solved by an analytical procedure. The Block-Pulse functions as basic functions are proposed to solve these equations. In order to investigate the validity of this method, we used the Adomian decomposition method to approximate the solution of the stochastic Du ng equations. The results reveal that the proposed method is very e ective.

At the present time the technical achievements of experimental studies make it possible to obtain the ions heavy elements with one electron, so electromagnetic interaction becomes the order of one, so that electromagnetic interactions become strong and the calculation of relativistic corrections becomes necessary. However, the theoretical models, intended for describing the relativistic corrections to the spectrum, are limited to the lowest order on the coupling constant. We will consider this problem, according to the asymptotic behaviour of the loop function in the scalar electrodynamics field and use the oscillator representation method. To make more sense of the calculations, mesonic hydrogen atom system has been studied to pave calculation methods for other atoms and systems including quarks, glueball, and pomeron which can be over- generalized using the intended potential.

All oscillators are periodically time varying systems, so to accurate phase noise calculation and simulation, time varying model should be considered. Phase noise is an important characteristic of oscillator design. It defined as the spectral density of the oscillator spectrum at an offset from the center frequency of the oscillator relative to the power of the oscillator. In this paper, we study linear time invariant (LTI) and linear time variant (LTV) model’s to calculate phase noise. Moreover, we propose a simple method for Impulse Sensitivity Function (ISF) calculation. Different oscillators have been selected to evaluate the proposed method. Simulation results show that the proposed method is simpler than other methods, and we can easily simulation ISF.