Analyzing the nonlinear vibration of beams is one of the important issues in structural engineering. According to this, an impressive analytical method which is called modified iteration perturbation method (MIPM) is used to obtain the behavior and frequency of a cantilever beam with geometric nonlinear. This new method is combined by the Mickens and iteration methods. Moreover, this method does not require small parameter in the equation which is difficult to be found for nonlinear oscillation. The accuracy of the solution that is obtained by the use of MIPM has been shown graphically and compared with exact solution. Comparison shows that good adaptation is obtained and MIPM as a powerful method for solving the vibrational behavior of structures analytically.

Introduction: Exchange rate is a measure of the equality of a country's currency against the currencies of other countries. It indicates the measurement of that country's economic situation in comparison with other countries. In the framework of conventional economic theory, the exchange rate system refers to the mechanism of determining the exchange rate through market forces exerted on the supply and demand. The purpose of this study is to understand the dynamics governing the exchange rate behavior using nonlinear models. By understanding the exchange rate dynamics, one can recognize its unusual and worrying behavior over time and apply the necessary policies accordingly. Methodology: The baseline linear model used in this study is a finite-order autoregressive (AR) model with relation (1): (1) In the real logarithm of the real exchange rate, the interrupted polynomials are placed in the roots of ϕ (z) = 0 on or outside a single circle. The roots outside the unit circle mean that PPP remains stable in the long run. Regime change regression Two-mode TAR (2) models are defined as follows: (2) In the above relation, there is a sequence of white noises with mean zero and variance 1. In this model, it is assumed that the variance in each regime is different from that in the other regimes. In order to complete the above definition, the R1 and R2 regimes need to be described more precisely. It depends on how each regime changes over time. Introducing the Bayesian model In order to estimate the Bayesian SETAR, we assume that the exchange rate variable has a normal distribution. (3) In the SETAR model, the regime change is defined as a discrete variable as follows: In the next step, in order to have a Bayesian estimate, we need to specify the backgrounds of the model coefficients and the other parameters. An appropriate assumption in this regard is to assume that the anterior distribution r is a continuous uniform distribution whose boundaries include the minimum and maximum time series data as follows: In the next step, we will define the backgrounds for the coefficients. According to the objectives of this study, we will use the background for the ignorance of the normal part as follows: Bayesian estimation method The basis of Bayesian inferences is Bayesian theorem. According to this theorem, the posterior probability of an event varies according to the product of the previous probability in the logarithm of the orthogonality. In mathematical terms, Bayes' theorem is as follows: Results and Discussion: Table 1 reports the results of the SETAR model estimation for the exchange rate return (Rials against the dollar with the monthly rotation in the period from 2004 to December 2020). The validity intervals of the coefficients are adjusted and do not include zero, which, like the classical case, is based on the significance of these coefficients. Table 1. Bayesian SETAR model coefficients (1) for the dollar exchange rate Coefficients Posterior average Posterior standard deviation 95% confidence interval 0, 0079 0, 0023 (0, 0034, 0, 0123) 0, 3179 0, 0713 (0, 1773, 0, 4567) 0, 0147 0, 0161 (-0, 0172, 0, 0465) 0, 4338 0, 1468 (0, 1427, 0, 7244) 0, 0223 0. 0006 (0, 0208, 0, 0230) Table 2. Variances of the two regimes in the Bayesian SETAR (1) model for the dollar exchange rate Coefficients Posterior average Posterior standard deviation 95% confidence interval 0/0007 0/0001 (0, 0006, 0, 0009) 0/0094 0/0019 (0, 0064, 0, 0137) According to the results of Table (2), when the exchange rate is lower than the latter value of thresholds, its variability is much less than when the exchange rate is higher than the threshold value (variance in regime 2 is greater than that in regime 1) Figure (1) shows the autocorrelation of the simulated values in the latter estimation of the model parameters in the two regimes. Figure 1. Autocorrelation of the posterior coefficients in both regimes 1 and The results in Figure 1 show that the correlation of the simulated values for all the model parameters rapidly decreases to zero. Therefore, we are faced with a suitable sample of values to simulate the posterior distribution of the parameters. There is also no need to increase the simulation volume. Figure (2) shows the effect curves of all the later parameters of the model used in this research: Figure 2. Effect diagrams for the SETAR pattern parameters (1) Based on the findings about the curve of the effect related to all the parameters, no regular pattern exists in the simulated values ​​of the parameters. Therefore, the stability of these coefficients is confirmed, and the results of the Bayesian model SETAR (1) used in this study are statistically valid. Conclusion: The exchange rate as a price variable plays a very important role in the performance of an economy. The results of this study indicate that there are two exchange regimes in which the exchange rate adjustment parameter will be in equilibrium with a 95% probability in regime 1 (0. 1773, 0. 4567). This is very small due to the deviation of the latter standard of this coefficient (S. Dev = 0. 0713). Also, the same parameter with a 95% probability in regime 2 will be at the distance (0. 1427, 0. 7244), which is a relatively long distance. The results showed that this is due to the high volatility of the exchange rate in regime 2. In addition, the adjustment in the first regime to the long-term equilibrium path is much safer than the adjustment to the long-term path in the second regime because the variability in the conditions of the exchange rate increase is very high. Finally, the results of this study showed that the expansionary exchange rate regime (regime 1) has a deviation from higher regime standards than the mild exchange rate regime (regime 2), which indicates high currency fluctuations in this regime and more uncertainty. So, using regime 1 provides the conditions for a proper economic growth in the future.

In this paper, the continuse Legendre wavelets constructed on the interval [0, 1] are used to solve the nonlinear Fredholm integro-differential equation. The nonlinear part of integro-differential is ap- proximated by Legendre wavelets, and the nonlinear integro-differential is reduced to a system of nonlinear equations. We give some numerical examples to show applicability of the proposed method.

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.

This paper is devoted to prove the existence of extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputo type and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. As an application, some examples are presented to illustrate the main results.

In this paper, nonlinear vibration control of a plate is investigated using a nonlinear modified positive position feedback method that is applied through a piezoelectric layer on the plate. Based on the classical theory of displacement and strain relations with von Karman, intended equations of motion for the smart plate have been obtained. In this model, transverse vibrations are studied and stimulations are performed for the primary resonance. Boundary conditions of the smart plate are simply supported. The plate is thickness symmetrical. Using the Galerkin method the temporal nonlinear equations governing the system have been found. Then, the free and forced vibrations of the structure with the nonlinear modified positive position feedback controller have been solved using the Method of Multiple Scales to obtain an analytical solution. Results show that this controller reduces the amplitude of the vibration by inducing an increase in the damping coefficient. In addition, this provides a higher level of suppression in the overall frequency domain response by increasing the compensator gain. Finally, the results of the analytical solution for the closed-loop nonlinear modified positive position feedback controller are presented and compared with the result of the conventional positive position feedback controller and nonlinear integral resonant controller. The results show that the performance of the nonlinear modified positive position feedback controller is better than other controllers and significantly reduces the vibration amplitude.

In this paper, an optimal robust nonlinear control approach based on the state-dependent Riccati equation (SDRE) is presented for a class of fractional-order nonlinear systems. The SDRE control is a method for the solution of nonlinear systems optimal problems, which has some disadvantages, such as it is not robust against disturbances and uncertainties and has not finite time convergence ability. Sliding mode control (SMC) is also a common method for nonlinear systems control, although it is not an optimal control method, it has great advantages such as robustness which has made it very popular. In the proposed method, a combination of SMC and SDRE methods is used for optimal and robust nonlinear control, which is suitable for a class of fractional-order nonlinear systems. The proposed control method has a robust structure against mismatched disturbances. The closed loop system stability is guaranteed by the Lyapunov theory and simulation examples show the performance improvement of the proposed method.

Ever-increasing demands of the modern world and the growth of industry requirements toward more accurate analyses have made the engineering community develop the first fundamental step and meet the needs by extending the previous prevalent linear methods into nonlinear areas. In this regard, the improvement of linear modes as one of the most pervasive and widespread analytical methods opens a new window to analyses with more closeness to reality. In this paper, after the deep identification of nonlinear Normal Modes, an approach is proposed to analyze the multidegree-of-freedom structures with nonlinear material under undamped free vibration. Afterward, through an in-depth investigation of the calculation methods, a novel algorithm for identifying nonlinear Normal Modes was proposed, and by expanding this algorithm to the existing invariable motion equation used in free vibration analysis, the possibility of the extraction of all nonlinear Normal Modes has emerged. After that, to investigate the functionality of the proposed approach, the Finite Elements Method-based Model of a 2-story steel structure was developed and, after verification, it was used to form the invariable differential equations. Finally, after verifying the Independent Periodic Method, pseudo-continuous masses of nonlinear Normal Modes and FrequencyEnergy curves of the mentioned structure were calculated. It is worth noting that the independency of resulted response to previous points, the possibility of capturing nonlinear Normal Modes with different frequencies in each degree-of-freedom, the potential of capturing all internal resonances, the expendability of Finite Elements Model to a set of invariable motion equations, and considering material nonlinearity are among achievements of the current paper.

In this paper, we first introduce the concepts of an equilibrium region and a stable equilibrium region. Then, we state and prove conditions under which a nonlinear autonomous system has a stable equilibrium region. It will be shown that these concepts and stability conditions are generalizations of the corresponding concepts of equilibrium points and asymptotical stable equilibrium points and their stability conditions. Using these conditions and Poincare-Bendixson theorem, we discuss conditions for the existence of periodic solutions in a nonlinear system of ordinary differential equations.