American Journal of Mechanical Engineering, 2013, Vol. 1, No. 3, 58-65 Available online at http://pubs.sciepub.com/ajme/1/3/1 © Science and Education Publishing DOI:10.12691/ajme-1-3-1 An Overview of the Mechanics of Oscillating Mechanisms Christopher G. Provatidis * Department of Mechanical Engineering, National Technical University of Athens, Greece *Corresponding author: cprovat@central.ntua.gr Received December 31, 2012; Revised April 26, 2013; Accepted May 09, 2013 Abstract This paper extends a previous study on the mechanics of oscillating mechanisms in which motion of an object is produced by attached rotating eccentric masses. In addition to the well known twin contra-rotating pair (Dean drive), the single eccentric mechanism is studied. In contrast to the contra-rotating systems where the initially still object moves along a vertical track, this study shows that in case of a single eccentric mass the track of the same object is extended in both the horizontal and vertical directions. In both configurations, the path is mainly influenced by the initial linear and angular momentum of the eccentrics. While contra-rotation requires a motor to ensure constant angular velocity, in single systems the latter event is conditionally achieved per se. In order to demonstrate the significance of the initial angular momentum in the motion of the object, the two types of eccentrics were applied for the conditions of the elementary Rutherford-Bohr’s model of a hydrogen atom and also of a virtual hydrogen molecule. It was found that, if intermolecular forces suddenly disappear at a specific synchronization, the virtual molecule or atom is predicted to reach the incredible altitude of 72km. Keywords: Two-body motion, synchronization, inertial propulsion, dean drive, hydrogen atom 1. Introduction Inertial propulsion appears in nature, mainly for navigation in diptera [1], and in industrial applications like shakers in mobile phones, wash-machines and many others [2]. The possibility to convert rotary motion into unidirectional motion has been presented by Dean [3] in mid-fifties but only recently details have been reported for the particular case of twin contra-rotating eccentrics attached to the object we wish to move [4,5]. The main conclusion was that motion is controlled mainly by the initial position 0  of the rotating masses when the object is left to fall down into the air, and also by the magnitude of the lumped masses m as well as by the angular velocity ω and the radius r (the product mωr constitutes the “angular momentum” of the propulsion system). When the motor keeps on working, the angular velocity may be controlled to be preserved at a constant value, while when it is switched off the conservation of mechanical energy demands the periodical variation of the angular velocity [5]. In addition to the abovementioned twin eccentrics, there are many cases where only one eccentric mass is attached to an object. Typical examples are: twin mechanisms suddenly broken, shakers in mobiles where the friction plays a significant role, athletes who rotate their arms during long jump, electrons that rotate around protons in atoms, and many others. This paper aims at studying primarily the single eccentric mechanism and also to compare it with the twin one. 2. Inertial Propulsion Mechanisms 2.1. Contra-rotation (Twin Eccentricity) 2.1.1. General Formulation The particular case of contra-rotating and synchronized eccentric masses has been previously investigated [4,5]. We consider the particular mechanism that consists of a horizontal object (rigid plate) characterized by a large mass M, on which two rigid rods of equal length (radius r) are symmetrically articulated at their one end while a smaller mass m is attached to their other ends, as shown in Figure 1. Details are given elsewhere [5], but for reasons of completeness we must make clear that: ● The object of mass M is rigid and has a uniform shape in the horizontal x-direction. ● The eccentric masses (No.1 and No.2) rotate at equal and opposite angular velocities,   t t     : contra- rotation. ● Both masses (No. 1 and No. 2) possess the same vertical z-level. ● The object is left free to fall down when the orientation of the rods form an angle 0  with the horizontal line, as shown in Figure 1. Obviously, the above ideal symmetric conditions lead to an ideal upward translational motion of the mechanism. The analysis below is based on the use of Lagrange equations.