REACTANCES AND SUSCEPTANCES OF MECHANICAL SYSTEMS

The classical solution to the problems associated with calculating the velocities and reactions of elements of complex mechanical systems under harmonic force consists in the compilation and integration of systems of differential equations and is rather cumbersome and time-consuming. In most cases, a steady state is of major interest. The purpose of this study is to develop essentially compact methods for calculating systems under steady-state conditions. The problem is solved by the methods which are typically used to calculate electrical circuits. Representation of harmonic quantities as rotating vectors in a complex plane and the operations with their complex amplitudes can greatly facilitate the calculation of arbitrarily complex mechanical systems under harmonic effects in the steady state. In the proposed method, a key role is played by mechanical reactance, resistance, and impedance for the parallel connection of consumers of mechanical power, as well as susceptance, conductance, and admittance for the serial one. At force resonance, the total reactance of the mechanical system is zero. This means that the system does not exhibit reactive resistance to the external harmonic force. At velocity resonance, the total susceptibility of the mechanical system is zero. This means that the system has infinitely high resistance to the external harmonic force. As a result, the stock of the source of harmonic force is stationary, although the inert body and the elastic element oscillate.

Energetic ion depletions near Europa and Io: the effect of plumes and atmospheric charge exchange

<p><strong>Introduction</strong></p><p>The flux of energetic ions (protons, oxygen and sulfur) near the Galilean moons were measured by the Energetic Particle Detector (EPD) on the Galileo mission (1995 - 2003). Near Galilean moons (such as Io and Europa) depletions of the energetic ion flux, of several orders of magnitude, were identified.</p><p>Such energetic ion depletions can be caused by the absorption of these particles onto the moon’s surfaces or by the loss due to charge exchange with neutral molecules in the atmospheres or potential plumes. To interpret the depletion features in the EPD data, a Monte Carlo particle tracing simulation has been conducted. The expected fluxes of the energetic ions are simulated under different scenarios including those with and without an atmosphere or plume. By comparing the simulated flux [YF1] to the EPD data, we investigate the cause of the depletion features with particular focuses on Europa and Io flybys.</p><p><strong>Results</strong></p><p>For Europa we report the following findings:</p><ul><li>For flyby E12 we find that a global atmosphere should produce a depletion region along the trajectory that is symmetrical to the closest approach, for energetic protons in the energy range of 80-220 keV. No such feature is visible in the data. Upper limits of the atmosphere are consistent with surface densities (⩽ 10<sup>8 </sup>cm<sup>-3</sup>) and scale heights (50-350 km) of previous studies. We find that a depletion of energetic protons (80-220 keV) occurring before closest approach is consistent with the field perturbations associated with a plume. This plume features coincides in time with the plume reported by Jia et al., 2018.</li> <li>For flyby E26 we find that the depletions of energetic protons (80-220 keV) are consistent with a simulation that takes into account the perturbations of the fields as calculated by an MHD simulation and atmospheric charge exchange. Furthermore, a depletion feature occurring shortly after closest approach is consistent with the field perturbations associated with a plume, located near the plume reported by Arnold et al., 2019.</li> <li>From these investigations, we confirm, independently from previous reports, that the Galileo spacecraft could have passed near plumes.</li> </ul><p>For Io we report the following results:</p><ul><li>We identify regions of proton (80-220 keV) depletions during Io flybys I24, I27 and I31 extending beyond one Io radius. The depletions features are not consistent with Io as an inert body. We investigate atmospheric charge exchange as a cause for the depletions.</li> </ul>

Anti-Resonance — Velocity Resonance

The task of the study is to establish the nature of mechanical resonance, namely, it is a resonance of forces or speeds. Two definitions are introduced. Definition 1. Resonance of forces is a resonance arising at a frequency ω = (k/m)0,5 in a mechanical system including an inert body and an elastic element, at which the reactive forces developed by them are maximal and opposite. Definition 2. The velocity resonance is a resonance arising at a frequency ω = (k/m)0,5 in a mechanical system, including an inert body and an elastic element, at which the speeds developed by them are maximum and opposite. The equation of forced mechanical oscillations corresponds to a parallel connection scheme, in which the inert body and changes in the dimensions of the elastic element and damper have a uniform speed, and their reactive forces are added. The sum of the reactive forces of the consumers of mechanical power is equal to the force developed by the source of mechanical power, which, like a voltage source in electrical engineering, can be called a source of power. Theorem 1 holds. If the condition ω = (k/m)0,5 is satisfied in a mechanical system consisting of parallel-connected inert bodies, an elastic element and a damper, a resonance of forces occurs. The inert body, the elastic element and the damper can be connected not only in parallel but also in series. With a series connection, a single force is applied to the elements of the system, and the velocities of the inert body and the changes in the dimensions of the elastic element and damper are added. The sum of the speeds of consumers of mechanical power is equal to the speed developed by the source of mechanical power, which, like a current source in electrical engineering, can be called a source of speed. Theorem 2 is valid. Under the condition ω = (k/m)0,5 in a mechanical system consisting of a series-connected inert body, an elastic element and a damper, a velocity resonance occurs. The mechanical resonance described in the courses of theoretical mechanics is the resonance of forces. It corresponds to a parallel connection of an inert body, an elastic element and a damper. When these elements are connected in series, a velocity resonance occurs.