A Note on the Mechanics of Ancient Gear Systems

This paper deals with the mechanical behavior of the gearwheels of the antiquity, which were generally characterized by triangular shaping of the teeth. The engagement of the conjugate profiles is analyzed in detail, calculating the temporal variation of the speed ratio due to the back and forth shifting of the relative instant center. The admissibility of the points of the theoretical contact path is carefully checked, estimating also the magnitude of the successive tooth collisions and ascertaining the energy losses arising from the particular nature of the coupling. Some very interesting results are that only one couple of teeth turns out to be active at each time instant and that the real path may belong to the only approach region or to the only recess region entirely or may be split into two separate sub-phases, the one in approach and the other in recess, or may even straddle both regions. The occurrence of each of these conditions depends on the average speed ratio (tooth ratio) and on the assigned clearance between the two wheels. It is also found that the speed oscillation is roughly contained in a ±10% range and the efficiency may reach rather high values, despite the presumable crude finishing of the ancient gearwheels due to the rudimentary technology used in the construction of the tooth profiles

On the Solution of Markov-switching Rational Expectations Models

This paper describes a method for solving a class of forward-looking Markov-switching Rational Expectations models under noisy measurement, by specifying the unobservable expectations component as a general-measurable function of the observable states of the system, to be determined optimally via stochastic control and filtering theory. Solution existence is proved by setting this function to the regime-dependent feedback control minimizing the mean-square deviation of the equilibrium path from the corresponding perfect-foresight autoregressive Markov jump state motion. As the exact expression of the conditional (rational) expectations term is derived both in finite and infinite horizon model formulations, no (asymptotic) stationarity assumptions are needed to solve forward the system, for only initial values knowledge is required. A simple sufficient condition for the mean-square stability of the obtained rational expectations equilibrium is also provided.Rational Expectations, Markov-switching dynamic systems, Dynamic programming, Time-varying Kalman filter

Preventing the oil film instability in rotor-dynamics

Horizontal rotor systems on lubricated journal bearings may incur instability risks depending on the load and the angular speed. The instability is associated with the asymmetry of the stiffness matrix of the bearings around the equilibrium position, in like manner as the internal hysteretic instability somehow, where some beneficial effect is indeed obtainable by an anisotropic configuration of the support stiffness. Hence, the idea of the present analysis is to check if similar advantages are also obtainable towards the oil film instability. The instability thresholds are calculated by usual methods, such as the Routh criterion or the direct search for the system eigenvalues. The results indicate that the rotor performances may be improved in the range of low Sommerfeld numbers by softening the support stiffness in the vertical plane, and hardening it on the horizontal one, up to the complete locking, though this advantage has to be paid by rather lower instability thresholds for large Sommerfeld numbers. Nevertheless, a "two-mode" arrangement is conceivable, with some vertical flexibility of the supports for large journal eccentricity, and complete locking for small eccentricity. As another alternative, the support anisotropy may be associated with the use of step bearings, whose particular characteristic is to improve the stability for small eccentricities

On the gear mechanics before the cycloid and involute profiling

After the extraordinary development of the machinery in the Hellenistic antiquity, the gear technique was transmitted to the Middle Ages through the Byzantine and Islamic culture and then to the Modern Era. The tooth profile was very crude, often trapezoidal or even rectangular and the gear behavior differed substantially from the modern involute profile. The kinematics of trapezoid profiles is here analyzed in detail, focusing on the temporal variation of the speed ratio due to the back and forth shifting of the relative instant center. Considering an isolated tooth pair, an approach phase is firstly observable, where the tip of the driven profile is pushed by the driver flank and then, after passing through the matching configuration, a recess phase follows, where the tip of the driver profile pushes the flank of the driven one. The acceptability of each configuration of this theoretical evolution is then checked according to the interference prevention requirement for the following and preceding tooth pairs and for the back inactive profiles. This yields some limitation to the tooth thickness, which may justify the frequent tooth slenderness in the design of the gear systems of the Renaissance. The periodic tooth collisions due to the jerky variability of the speed ratio are properly analyzed, together with the energy losses arising from the sliding friction. Overall, the results show that only one tooth pair is active at each time instant and the acceptable contacts may belong to the only approach region, or to the only recess region, or may be split into two separate sub-phases, in approach and in recess, or may even straddle both regions. The occurrence of each of these situations depends on the average speed ratio and on the assigned clearance between the two wheels

A Two-Dimensional Approach to Rubber V-Belt Mechanics

A new two-dimensional approach to the mechanics of rubber V-belt CVT's takes into account the change, along the belt sides, of the sliding velocity on the pulley walls, together with the cross section rotation due to the shear forces. The results show significant differences with the one-dimensional thin belt model and point out the gradual increase or decrease of the belt curvature in the entrance and exit regions of the contact arc, which implies the negligibility of the belt arching in the free spans. Some experimental tests on the global variables of a rubber belt CVT indicate a very fine acceptability of the theoretical model

Tidal and nonequilibrium Casimir effects in free fall

In this work, we consider a Casimir apparatus that is put into free fall (e.g., falling into a black hole). Working in 1 + 1D, we find that two main effects occur: First, the Casimir energy density experiences a tidal effect where negative energy is pushed toward the plates and the resulting force experienced by the plates is increased. Second, the process of falling is inherently nonequilibrium and we treat it as such, demonstrating that the Casimir energy density moves back and forth between the plates after being “dropped,” with the force modulating in synchrony. In this way, the Casimir energy behaves as a classical liquid might, putting (negative) pressure on the walls as it moves about in its container. In particular, we consider this in the context of a black hole and the multiple vacua that can be achieved outside of the apparatus

The Rotary Aero-Engine from 1908 to 1918

The rotary aero engine is a special type of air-cooled radial engine, where the cylinders are arranged like the spokes of a wheel and turn around the crankshaft. The propeller is connected to the cylinders, while the crankshaft is fixed to the frame. The rotary aero engine, developed in 1908, set new standards of power and light weight within the aircraft industry. It was adopted by many pioneer aviators and widely used to set records of endurance, speed and height. Many aero engine manufacturers produced different models and variants of this type of engine, which was extensively used until the end of the First World War. The latest evolution of the rotary engine was the counter-rotary arrangement, which was devised and designed by the Siemens-Halske company. The distinctive feature of this type of engine was that the engine body (with cylinders and propeller) rotated in one direction while the crankshaft rotated in the opposite one. This result was obtained by using a bevel gear mechanism. However, rotaries were quickly and definitively replaced in 1918 by new kinds of conventional engine, which were developed in the same period by other manufacturers. The main features of rotary and counter-rotary aero engine and the performance limits that caused their decline will be described in this paper. The rotary engine will be compared with the conventional one in terms of power output, specific consumption, weight and inertia loads transferred to the frame

Casimir effect in free fall towards a Schwarzschild black hole

In this paper we discuss the Casimir effect in a small cavity, freely falling from spatial infinity in spacetime geometry outside of a Schwarzschild black hole. Our main goal is to search for possible changes in the vacuum energy, as well as particle creation inside the falling cavity, with respect to a comoving observer. Working in the Lemaître chart and assuming a cavity size L much smaller than the Schwarzschild radius (L/r_g≪1), we solve the Klein-Gordon equation for a massless scalar field confined within the cavity in the reference frame of the comoving observer. We follow Schwinger’s proper time approach, evaluating the one-loop effective action for the field in the falling cavity hence evaluating the corrections to the vacuum energy. We find a small reduction in the absolute value of Casimir energy as the cavity approaches the black hole horizon due to the changing spacetime geometry. Since the spacetime geometry for the cavity changes dynamically, we further find the energy density of the created particles due to the dynamical Casimir effect. These dynamical contributions exactly match the deficit to the static Casimir energy. Combined, the observer measures a net increase in energy within the cavity as she falls