Periodic oscillations for a graphene-based electrostatic micro actuator

We study the mechanical oscillations for a novel model of a graphene-based electrostatic parallel plates micro actuator introduced by Wei et al.(2017), considering damping effects when a periodic voltage with alternating current is applied. Our analysis starts from recent results about this MEMS model with constant voltage, and provides new insights on the periodic mechanical responses for a variable input voltage. We derive sufficient conditions on the system physical components for which periodic oscillations with constant sign exist together with their stability properties. Specifically, under some conditions, the existence of three periodic solutions is established, one of them is negative and the others are positive in sign. The positive one nearby the origin is asymptotically locally stable, whilst the other two are unstable. Additionally, we prove that no further constant sign periodic solutions can be found. The existence of periodic solutions is approached from direct and reverse order Lower and Upper Solutions Method, and the stability assertions are derived from the Liapounoff-Zukovskii criteria for Hill's equations and the linearization principle. Theoretical results are complemented by numerical simulations and numerical continuation results. Furthermore, these numerical simulations evidence the robustness of the graphene-based MEMS model over the traditional ones.

A new perturbative solution to the motion around triangular Lagrangian points in the elliptic restricted three-body problem

The equations of motion of the planar elliptic restricted three-body problem are transformed to four decoupled Hill’s equations. By using the Floquet theorem, a perturbative solution to the oscillator equations with time-dependent periodic coefficients are presented. We clarify the transformation details that provide the applicability of the method. The form of newly derived equations inherently comprises the stability boundaries around the triangular Lagrangian points. The analytic approach is valid for system parameters $$0 < e \le 0.05$$ 0 < e ≤ 0.05 and $$0 < \mu \le 0.01$$ 0 < μ ≤ 0.01 where e denotes the eccentricity of the primaries, while $$\mu $$ μ is the mass parameter. Possible application to known extrasolar planetary systems is also demonstrated.

Geometry and complexity of path integrals in inhomogeneous CFTs

In this work we develop the path integral optimization in a class of inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a position dependent metric. After setting up and solving the general optimization problem, we study specific examples, including the Möbius, SSD and Rainbow deformed CFTs, and analyze path integral geometries and complexity for universal classes of states in these models. We find that metrics for optimal path integrals coincide with particular slices of AdS3 geometries, on which Einstein’s equations are equivalent to the condition for minimal path integral complexity. We also find that while leading divergences of path integral complexity remain unchanged, constant contributions are modified in a universal, position dependent manner. Moreover, we analyze entanglement entropies in inhomogeneous CFTs and show that they satisfy Hill’s equations, which can be used to extract the energy density consistent with the first law of entanglement. Our findings not only support comparisons between slices of bulk spacetimes and circuits of path integrations, but also demonstrate that path integral geometries and complexity serve as a powerful tool for understanding the interesting physics of inhomogeneous systems.

Stability of an exponentially tapered asymmetric sandwich beam placing on a variable Pasternak Foundation with variable temperature gradient

The stability investigation of an exponentially tapered sandwich beam, asymmetric in nature placed upon a Pasternak foundation with variable behavior acted upon by a periodic longitudinal load with variable temperature grade with clamped-pinned condition provided at the ends is analyzed in this article. By using Hamilton’s energy method, a complete solution for the mathematical modeling of the system is obtained. The equations of motion along with the related boundary conditions are obtained in non-dimensional form. A group of Hill’s equations are found by generalized Galerkin’s method. Different parameters have significant influence on both the static buckling loads as well as the zones of instability. These effects of these parameters are examined and are presented in a graphical manner. The outcomes resulted due to uniform and variable temperature grade are compared.

Ulam Stability for a Class of Hill’s Equations

This paper deals with Ulam’s type stability for a class of Hill’s equations. In the two assertions of the main theorem, we obtain Ulam stability constants that are symmetrical to each other. By combining the obtained results, a necessary and sufficient condition for Ulam stability of a Hill’s equation is established. The results are generalized to nonhomogeneous Hill’s equations, and then application examples are presented. In particular, it is shown that if the approximate solution is unbounded, then there is an unbounded exact solution.

Impulse control flight to the invariant manifold near collinear libration point

We consider the special problem of flight from near-Earth orbit to a neighborhood of first collinear libration point of the Sun-Earth system. For such flight the numerical experiments substantiate the adequacy of the model of Hill’s equations, which is the nonlinear approximation of equations of circular limited three-body problem. Otherwise, we would be obliged to use the model of limited three-body problem (or its approximation) in conjunction with the model of two-body problem for modeling of motion. During of approach to the neighborhood of libration point (in space of positions), the series of impulse controls are implemented. Controls are built on the basis of equations in variations. The purpose of implementing presented controls is hitting the manifold, where a spacecraft will be as long as possible in the linear case. This manifold is achieved when the special functions of phase variables is equal to zero. All the presented studies are illustrated in detail.

Static and Dynamic Stability Analysis of a Rotating Taper Beam Having Elliptical Cross Section Subjected to Pulsating Axial Load With Thermal Gradient

The static and dynamic stability of a rotating tapered beam having an elliptical cross-section subjected to a pulsating axial load with a thermal gradient is investigated under three different boundary conditions, such as clampedclamped (C-C), clamped-pinned (C-P), and pinned-pinned (P-P). The governing equations of motion have been derived by using Hamilton’s energy principle. A set of Hill’s equations have been obtained by the application of generalized Galerkin’s method. The effects of taper parameter, hub radius, rotational speed, thermal gradient, and geometric parameter on the static buckling loads and the regions of instability have been studied and the results are presented graphically