We present the first analytical superposition of a charged black hole with an annular disk of extremal dust. In order to obtain the solutions, we first solve the Einstein-Maxwell field equations for sources that represent disk-like configurations of matter in confomastatic spacetimes by assuming a functional dependence among the metric function, the electric potential and an auxiliary function, which is taken as a solution of the Laplace equation. We then employ the Lord Kelvin Inversion M ethod applied to models of finite extension in order to obtain annular disks. The structures obtained extend to infinity, but their total masses are finite and all the energy conditions are satisfied. Finally, we observe that the extremal Reissner-Nordström black hole can be embedded into the center of the disks by adding a boundary term in the inversion.
The frictional force between a physical damped pendulum and the medium is usually assumed to be proportional to the pendulum velocity. In this work, we investigate how the pendulum motion will be affected when the drag force is modeled using power-laws bigger than the usual 1 or 2, and we will show that such assumption leads to contradictions with experimental observations. For this purpose, a more general model of a damped pendulum is introduced, assuming a power-law with integer exponents in the damping term of the equation of motion, and also in the non-harmonic regime. A Runge–Kutta solver is implemented to compute the numerical solutions for the first five powers, showing that the linear drag has the fastest decay to rest, and that bigger exponents have long-time fluctuation around the equilibrium position, which have no correlation (as is expected) with experimental results.
This paper presents an alternate form for the dynamic modelling of a mechanical system that simulates in real life a gantry crane type, using Euler's classical mechanics and Lagrange formalism, which allows find the equations of motion that our model describe. Moreover, it has a basic model design system using the SolidWorks software, based on the material and dimensions of the model provides some physical variables necessary for modelling. In order to verify the theoretical results obtained, a contrast was made between solutions obtained by simulation in SimMechanics-Matlab and Euler-Lagrange equations system, has been solved through Matlab libraries for solving equation's systems of the type and order obtained. The force is determined, but not as exerted by the spring, as this will be the control variable. The objective is to bring the mass of the pendulum from one point to another with a specified distance without the oscillation from it, so that, the answer is overdamped. This article includes an analysis of PID control in which the equations of motion of Euler-Lagrange are rewritten in the state space, once there, they were implemented in Simulink to get the natural response of the system to a step input in F and then draw the desired trajectories.
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