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Introduction To Perturbation Techniques

Analytical Methods in Vibrations

The Chebyshev–Ritz route is exploited for the sake of the examination of the principal parametric resonance instability boundary of a rotating nanocomposite beam subjected to a temperature rising. The parametric resonance excitation is because of a time varying rotational speed. It is assumed that the rotating speed oscillates harmonically about a constant value. Resorting to the Halpin–Tsai micromechanical, the practical mechanical and thermal characteristics are estimated. The Newton–Raphson scheme is utilized to determine the prestressed configuration. The structural damping influence is incorporated to the motion equations. By the assumption that the frequency of the harmonically varying term of the rotating speed is approximately twice the one of the natural frequencies, the principal parametric resonance excitation is activated. The Bolotin’s technique alongside the Floquet theory are utilized to define the instability region boundaries. The impacts of various effectual parameters including the graphene platelet weight fraction, and its dispersion pattern, the design value of the damping constant, the boundary condition type, and the rotating speed on the instability region are examined. It is inferred that by the increment of the weight fraction of the graphene platelet as well as the design value of the damping ratio, the critical excitation amplitude increases. 

Application of the Chebyshev–Ritz route in determination of the dynamic instability region boundary for rotating nanocomposite beams reinforced with graphene platelet subjected to a temperature increment

A micro scale nonlinear beam model based on strain gradient elasticity is developed.Governing equations of motion and boundary conditions are obtained in a variational framework.As an example,the nonlinear vibration of microbeams is analyzed.In a beam having a thickness to length parameter ratio close to unity,the strain gradient effect on increasing the natural frequency is predominant.By increasing the beam thickness,this effect decreases and geometric nonlinearity plays the main role on increasing the natural frequency.For some specific ratios,both geometric nonlinearity and size effects have a significant role on increasing the natural frequency.

A nonlinear microbeam model based on strain gradient elasticity theory

The current manuscript deals with the fractional dynamical system of transmission along with co-infection of TB in the HIV community including ten classes. We study the necessary conditions for the existence and uniqueness of the solution of the considered system under the fractional operator known as Atangana–Baleanu in Caputo sense. By using fixed-point theory, the qualitative analysis of the result and Ulam–Hyers stability involving fractional operator for the proposed system is derived. For numerical simulations, we applied the fractional Adams–Bashforth method to the system. The obtained results are demonstrated explicitly to illustrate the validity of the considered technique for solving the proposed model under the above-mentioned operator. We observed that, if HIV-infected or unprotected individuals are in contact with TB infected individuals, the disease will grow in society. The dynamical behavior is shown for different arbitrary orders between 0 and 1, which converges to the integer-order one.

Investigation of mathematical model of transmission co-infection TB in HIV community with a non-singular kernel

In this study, the dynamic bifurcation of a viscoelastic micro rotating shaft is investigated. The non-classical theory (the modified couple stress theory) and the Kelvin Voigt model are used for modeling the viscoelastic micro shaft. The transverse equations of motion are derived using the variational approach. The reduced order model of the system is obtained by the Galerkin method. Using the Routh–Hurwitz criteria the stability regions of the system are extracted in which the effect of the length scale parameter is significant. Using the center manifold theory and the normal form method the double zero eigenvalue bifurcation is analyzed. The results show that the internal and external damping coefficients, the rotational speed and the material length scale parameter influence the critical speed, amplitude, and phase of a non-trivial solution, and radius of limit cycle (periodic solution). Also, it is seen that by increasing the dimensionless length scale parameter (material length scale per radius of the shaft) the radius of the limit cycle is decreased, whereas the critical rotational speed and the rate of the phase are increased. However, the radius of the limit cycle concerning the classical theory is higher than that of regarding the modified couple stress theory. Furthermore, with an increase of the external damping coefficient the radius of the limit cycle is linearly decreased; however, the critical speed of the system is increased. Additionally, by decreasing length scale parameter the results of the modified couple stress theory approach the classical theory ones.

Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this study the nonlinear dynamics of an asymmetrical rotating shaft is investigated. The high-static-low dynamic stiffness are used. Also, the active time-delayed proportional derivative control...

Time-delayed control of a nonlinear asymmetrical rotor near the major critical speed with flexible supports

In this paper, the free and harmonically forced vibrations of a viscoelastic rotating microbeam have been analyzed by exploiting non-classical theories, such as the Modified Couple Stress Theory and Modified Strain Gradient Theory (MSGT). A continuous model is derived and reduced to a well-known Duffing–Rayleigh system via a Galerkin projection; then, the Multiple Scales Method is used to obtain amplitude equations. Free and forced nonlinear oscillations are studied via classical and non-classical theories, and their corresponding results are compared with each other. An expression for natural frequency is obtained in which the effects of the internal damping and small-scale parameters are obvious. It is observed that the natural frequency obtained using the MSGT has the highest value, while the Classical Theory (CT) predicts the lowest natural frequency value. Also, it is seen that by increasing the hub radius, the influence of rotational speed on natural frequency is increased. According to obtained results, the responses corresponding to the CT become completely stable for the lowest value of the excitation force and the highest value of the external damping coefficient. For MSGT, responses are stable when the values of the excitation force and external damping coefficients are highest and lowest, respectively. It is shown that for the MSGT, the system is stable when the internal damping coefficient is small, while for the CT, the beam is stable for the highest value of the internal damping coefficient. The system parameters—such as the detuning parameter, damping coefficients, and excitation force—affect the solution, such a way that the system may possess one or two stable spirals depending on the values of those parameters, as shown in phase plane portraits.

Nonlinear dynamic behavior and bifurcation analysis of a rotating viscoelastic size-dependent beam based on non-classical theories

This paper deals with reduction of oscillations of a continuous spinning shaft using proportional-derivative time delay controller. The stable and unstable zones of the system are obtained using th...

S. Ali Ghasabi

Papers

Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

Time-delayed control of a nonlinear asymmetrical rotor near the major critical speed with flexible supports

Nonlinear dynamic behavior and bifurcation analysis of a rotating viscoelastic size-dependent beam based on non-classical theories

Analysis and suppression of the nonlinear oscillations of a continuous rotating shaft using an active time-delayed control