Abstract
This paper revisits the notorious Dean’s drive, aiming at causing unidirectional motion when attached to an object at rest (inertial propulsion). Instead of the usual circular path along which lumped eccentric masses move, a novel figure-eight-shaped curve is proposed. It is generally shown that the rotational kinetic energy accumulated in the rotating masses produces an equivalent linear ‘initial’ velocity to the object, and therefore, it further works like an oscillating projectile within the gravitational field. Conditions for the variation of the angular velocity, in order to achieve a ‘very short-time hovering’, are derived. The study does not consider any elastodynamic effects and reduces only to rigid-body analysis. Further improvements such as the spinning of the entire mechanism about its axis of symmetry are discussed.
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Abbreviations
- a :
-
Acceleration vector (m/s2)
- F :
-
Force (N)
- g :
-
Gravitational acceleration (m/s2)
- M :
-
Mass of the object (kg)
- m :
-
Mass attached to the end of each rotating rod (kg)
- R :
-
Half-distance between the two rods (m)
- r :
-
Radius of each rod (m)
- r :
-
Position vector
- T :
-
Rotation matrix (dimensionless)
- t :
-
Time (s)
- z c :
-
Position of the center of mass (m)
- z M :
-
Position of the object (m)
- ϕ :
-
Polar angle (rad)
- ω :
-
Angular velocity of contra-rotating masses (rad/s)
- ω z :
-
Angular velocity about the z axis (rad/s)
- a :
-
Relative to mass ‘a’
- b :
-
Relative to mass ‘b’
- z :
-
Toward the vertical axis
- 0:
-
Relative to initial time, t = 0
References
Blekhman II (1988) Synchronization in science and technology. ASME Press, New York
Blekhman II (2000) Vibrational mechanics: nonlinear dynamic effects, general approach, applications. World Scientific, Singapore, pp 472–497
Dean NL (1959) System for converting rotary motion into unidirectional motion. US Patent 2,886,976 (Filed Jul. 13, 1954, granted May 19, 1959). Also: http://en.wikipedia.org/wiki/Dean-drive, among others
Gardiner EN (2002) Athletics in the ancient world. Dover, New York
Goldstein H (1980) Classical mechanics, 2nd edn. Addison-Wesley, Reading
Goncharevich IF (1972) Dynamics of vibrational transportation. Nauka, Moscow, p 244
Kononenko VO (1969) Vibrating systems with a limited power supply. Ilife Books, London
Millis MG, Thomas NE (2006). Responding to mechanical antigravity. NASA/TM-2006-214390, AIAA-2006-4913, December 2006. Available at: http://gltrs.grc.nasa.gov/reports/2006/TM-2006-214390.pdf
Minetti AE, Ardigo LP (2002) Halteres used in ancient Olympic long jump. Nature 420:141–142
Provatidis CG (2009). A novel mechanism to produce figure-eight-shaped closed curves in the three-dimensional space. In: Tsahalis D (ed.), Proceedings of Third International Conference on Experiments/Process/System Modeling/Simulation & Optimization, Athens, 8–11 July 2009 ([CD-ROM])
Provatidis CG (2010) Some issues on inertia propulsion mechanisms using two contra-rotating masses. Theory Mech Mach 8(1):34–41
Provatidis CG (2010) A device that can produce net impulse using rotating masses. Engineering 2:648–657
Provatidis CG (2011) A study of the mechanics of an oscillating mechanism. Int J Mech 5(4):263–274
Provatidis CG (2013) Simplified biomechanics for a possible explanation of the ancient Greek long jump using halteres. Univers J Eng Sci 1(1):5–16
Provatidis CG and Gamble MA (2013). Support forces in a synchronized rotating spring-mass system and its electromagnetic equivalent. Int J Appl Electromag Mech, 41(3): 313–334. Also: Technical Report BOE063011-447, Released by Boeing Co., 25th June 2011; Also: The Boeing Co., Copyright 12-00613-EOT, Date: 16th August 2012
Stepanov GYu (1963) Why is it impossible to have ‘Dean’s Apparatus’? Jour Priroda 7:85–91 [in Russian]
Todeschini M (1933). Motore a forza propulsive centrifuga alimentato ad acqua dissociate con variazione di velocita graduale automatic. Regno d’Italia, Ministero delle Corporazioni, Brevetto Industriale No. 312496, 17 November 1933; see the first two pages of the patent at the website www.circolotodeschini.com
Vartholomeos P, Papadopoulos E (2008) Analysis and experiments of the force capabilities of centripetal-force-actuated microrobotic platforms. IEEE Trans Rob 24(3):588–599
Zhao C, Zhu H, Zhang Y (2010) Synchronization of two coupled exciters in a vibrating system of spatial motion. Acta Mech Sin 26:477–493
Zhao C, Zhang Y, Wen B (2010) Synchronisation and general dynamic symmetry of a vibrating system with two exciters rotating in opposite directions. Chin Phys B 19(3):030305-1-7
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I thank the anonymous reviewer for the thorough examination of the manuscript and his valuable comments, which substantially improved this paper.
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Appendices
Appendix 1: Balancing of lateral inertial forces when the object does not rotate around the z axis
In order to get a better idea about the inertial forces induced by the rotating masses, in this appendix we will consider that the object does not rotate around the z axis, i.e., ω z = 0. In this case, Eqs. (4) and (5) simplify to:
and
Therefore, the center of mass of the couple (a, b) will be at the position:
If m is the mass of either of particles ‘a’ or ‘b’, by virtue of (A-3), (A-4) and (A-5), the components of the resultant inertial force are given below:
The balancing of inertial force in the y-direction can be achieved using a second couple of masses (a′, b′) of the same characteristics and the same initial configuration, which rotates in the opposite direction at an angular velocity \(\omega^{{\prime }} = - \omega\) (i.e., \(\omega_{a}^{{\prime }} = - \omega_{a} = -\omega\) and \(\omega_{b}^{{\prime }} = - \omega_{b} = \omega\)). In such a case, concerning the ensemble of the two couples, (a, b) and (a′, b′), the total F x component remains equal to zero, the F y component becomes zero as an algebraic sum of two opposite quantities [\(\sum {F_{y} } = - 4m\omega^{2} r(\sin 2\omega t + \sin 2\omega^{{\prime }} t) \equiv 0\)], whereas the F z component becomes double than what it was when only the couple (a, b) existed.
In summary, the synergy of the couples (a, b) and (a′, b′) leads to the following resultant forces:
Appendix 2: Rotation about the z axis
Taking into consideration that \(\omega^{{\prime }} = - \omega\), the center of mass of the two masses (a′, b′) mentioned in Appendix 1 lies at the position:
Combining (A-4) with (B-1), the center of mass of the above-mentioned four masses (a,b) and (a′, b′), lies at the position:
Eq. (B-2) suggests that in the above perfect balance, the center of mass of the two devices (i.e., of the four masses) does not laterally move (X m = Y m = 0). Therefore, if we now consider the rotation ω z around the z axis, it is evident that this rotation will not further induce additional inertial forces. In other words, the object on which the two devices are attached will perform a purely translational motion in the z-direction.
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Provatidis, C.G. Unidirectional motion using rotating masses along figure-eight-shaped trajectories. J Braz. Soc. Mech. Sci. Eng. 37, 397–409 (2015). https://doi.org/10.1007/s40430-014-0154-y
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DOI: https://doi.org/10.1007/s40430-014-0154-y