Engineering, 2010, 2, 648-657
doi:10.4236/eng.2010.28083 Published Online August 2010 (http://www.SciRP.org/journal/eng).
A Device that can Produce Net Impulse Using Rotating
Masses
Christopher G. Provatidis
Laboratory of Dynamics & Structures, Mechanical Design & Control Systems Section, School of Mechanical
Engineering, National Technical University of Athens, Athens, Greece
Received March 20, 2010; revised July 6, 2010; accepted July 9, 2010
Abstract
This paper describes a device capable of producing net impulse, through two synchronized masses, which
move along a figure-eight-shaped orbit. In addition to the detailed description of the mechanical components
of this device, particular attention is paid to the theoretical treatment of the innovative principle on which the
device is based. In more details, the mechanical system consists of two independent but simultaneous rota-
tions, the former being related to the formation of the figure-eight-shaped path and the latter to an additional
spinning. Based on the parametric equations of motion of the lumped masses, and considering semi-static
tensile deformation of the connecting rods carrying them, it was found that the resultant impulse towards the
direction of the spin vector includes a non-vanishing term that is linearly proportional to the time. In addition,
reduced but encouraging experimental results are reported. These findings sustain the capability of the pro-
posed mechanism to achieve propulsion.
Keywords: Inertial Propulsion, Centrifugal Force, Net Impulse, Rotating Figure-Eight, Mechanism
1. Introduction
The matter of the inertial propulsion utilizing eccentric
masses aiming at producing limited motion of the object
to which they are attached, is an old topic [1-4]. The
general impression is that these masses lead to periodic
oscillations in which the synchronization of participating
masses plays a significant role [5], while chaotic phe-
nomena may also appear [6,7].
In the particular case of possible space propulsion,
fifty years ago Norman Dean (a civil service employee
residing in Washington DC) proposed the use of two
contra-rotating eccentric masses in order to convert ro-
tary motion to unidirectional motion [4]. He claimed that
in this way one could achieve thrust thus producing mo-
tion of the object to which this system was attached.
Since then, Dean’s mechanism was internationally nam-
ed as ‘Dean drive’ or ‘Dean space drive’ (the interested
reader may consult, for example, http://en.wikipedia.
org/wiki/Dean_drive). However, despite the extremely
high number of internet references as well as the many
articles sited in popular mechanics or science fiction ma-
gazines, a very small number of scientific papers exist in
the open literature. A careful search reveals an old paper
in the Russian language [8], two remarks in a textbook
[9], as well as a general NASA report dealing with many
possible mechanical antigravity concepts (including Dea-
n’s drive) [10] and a relevant review paper [11]. More-
over, quite recently Provatidis [12] has shown that Dea-
n’s drive practically works like a catapult while a vari-
able angular velocity can only control the smoothness of
the object velocity to which the drive is attached. In brief,
as it will be discussed below, the main disadvantage of
Dean’s drive is due to the circular paths on which the ec-
centric masses move.
Motivated by the abovementioned findings on Dean’s
drive [12], this paper revisits the subject and shows in a
theoretical way that, the shape of the path on which the
lumped masses move, is of major importance. For exam-
ple, if one considers a lumped mass at the end of an elas-
tic bar of radius r, which rotates at a constant angular
velocity

on a vertical plane, the mass moves along
an ideally circular path (C) because the induced cen-
trifugal force has a constant value thus leading to a per-
manent tensile elongation r of the rod (final radius:
rr r


 ). In this case, a rigid-body analysis is allow-
able, and also, due to the geometrical symmetry of the
path as well as due to the fact that every 360 degrees the

C. G. PROVATIDIS 649
mass takes exactly the same position possessing exactly
the same velocity, the net impulse caused by the cen-
trifugal force vanishes. In more details, when the mass
draws the upper half of the circle the corresponding im-
pulse is positive, while when it draws the lower half it
becomes negative (both of equal absolute value).
Within this context, this paper investigates the possi-
bility of strengthening the impulse on the upper part of
the circular path (appearing in Dean drive) with respect
to the lower one. A possible solution to this problem,
which has been previously presented [13,14] but it is
fully explained here for the first time, is to transform the
‘circle’ to a different shape. It is proposed to achieve it
by deforming the ‘circle’ in two successive ways. First
the ‘circle’ is folded by rotating its lower part around the
vertical axis of symmetry thus producing a crossed fig-
ure-eight-shape, which entirely lies on the vertical plane.
Second the latter planar path is further bent in such a way
that it perfectly lies over the surface of a hemisphere, the
latter having a center ‘O’ and a radius r. These two
successive deformation steps lead to a new, fully three-
dimensional, curvilinear path that lies entirely above or
entirely below the center of the hemisphere; henceforth it
is called ‘figure-eight-shaped’ path. It is clarified that in
this final configuration of the mechanism, the immobile
end of every connecting bar is pinned to the centre of the
hemisphere while the second end carries the correspond-
ing mass. Consequently, one could say that-in this way-
the proposed procedure achieves to create a new path on
which only the upper, or only the lower half of the ini-
tially considered circular path (C), operate. Despite this
fact, it has been theoretically verified that the maximum
upward force is equal and opposite to the maximum
downward force thus net propulsion is still impossible
[15]. A mechanical device capable of producing the afor-
ementioned figure-eight-shaped path is presented in Sec-
tion 2 and it is shown in Figure 1.
Figure 1. The prototype mechanism. In the left part the
arrows show the directions of the two simultaneous rota-
tions, while in the right part the figure-eight-shaped path is
clearly illustrated for two different views.
In order to increase the maximum upward force with
respect to the maximum downward one, the abovemen-
tioned figure-eight-shaped curve is further modified as
follows. Another rotation
z

is imposed around the
vertical axis of symmetry, which obviously passes thro-
ugh the center O of the hemisphere. Due to the additional
centrifugal forces ought to the rotation
z

, when the
elastic rod is found at the horizontal position is suffers
much larger tensile force than that it suffers at the verti-
cal position; therefore, the instantaneous elongated radius
at the horizontal position is higher than that of the verti-
cal position. As a result, the induced vertical inertial
forces - which are always proportional to the instantane-
ous radius r (cf. Equation (23)) - at the horizontal posi-
tion are higher than those corresponding to the vertical
position (a full explanation will be provided in Sections
5-7). In this way, the proposed mechanism that consists
of a rotating ‘figure-eight’ is capable of producing net
impulse, a finding that constitutes the novel feature of
this paper. The full description of the demonstration de-
vice, an experimental result, and particularly the theo-
retical treatment of the involved mechanics, are pre-
sented here for the first time.
The structure of this paper is as follows. Section 2 pre-
sents details of the proposed device concerning its con-
struction features, operation and design parameters. Then,
a theoretical treatment follows (Section 3 presents analy-
tical closed-form expressions of the figure-eight-shaped
path, Section 4 presents the velocities and the accelera-
tions of the lumped masses, Section 5 presents the ex-
pression for the resultant inertial forces, Section 6 deals
with the elastic deformation of the connecting bars and
Section 7 presents closed-form expressions for the resul-
tant impulse). Experimental results are presented in Sec-
tion 8, while a discussion follows in Section 9.
2. Presentation of the Innovative Device
2.1. Construction Features
An overview of the proposed device, which has been
developed so as to produce the abovementioned fig-
ure-eight-shaped path and its further spinning, is shown
in Figure 2. The mechanism consists of a frame (1) on
which two electric motors (M1, M2) are attached. The
mechanism consists of a horizontal shaft (2) supported
by the frame (1) guided by the electric motor (M1); in
this implementation through the shaft (7) and belt (8)
towards the end (6) of the shaft. The shaft (2) includes in
its interior other shafts that drive an attached planetary
system (3) that is positioned preferably in its middle.
Two rods (4,5) are attached in the ends of the spin gears’
axis of revolution (9) where masses (‘a’, ‘b’) are attached
and regularly move on the path. During the rotation of
the electric motor (M1), the masses (‘a’, ‘b’) move along

C. G. PROVATIDIS 650
a figure-eight-shaped path (shown in the right part of
Figure 1), which entirely belongs to the surface of a
sphere the center of which is the intersection of the hori-
zontal axis (2) with the axis of the spin gears (9); in other
words, the center of the planetic system (3). Finally, the
end of the shaft (2) includes a wheel (10), while the other
end is driven by the motor (M1); in this specific case
through the belt (8). Also, a second electric motor (M2)
is illustrated at the bottom of the frame (1), which causes
the rotation of the entire frame and consequently of the
shaft (2) as well as of the attached masses (‘a’, ‘b’).
The central part of the abovementioned device is the
planetary system. To make it more clearly to the reader,
Figure 3 is a cross-section of the horizontal shaft (2) and
the attached planetary system (3). The latter consists of a
casing (11) that expands almost until the ends of the
shaft (2). Internally, the planetary system (3) consists of
some planet gears and spin gears. Again, the case of two
planet gears (P1,P2) and two spin gears (S1,S2) is quite
indicative. The planet gears and the spin gears are sup-
ported on the casing through the rolling bearings
(211,212) and (213,214), respectively (212 and 214 are
not shown in Figure 3). In this case, the planet gear (P1)
is firmly fixed to the right half of the internal shaft (22)
that ends to the point (6), while the planet gear (P2) is
firmly fixed to the other half of the internal shaft (21)
that ends to the wheel (10). Finally, the rods (4,5) are
again shown together with the corresponding attached
masses (‘a’,’b’), which have been already mentioned in
Figure 2.
2.2. Operation
In brief, the motor M1 rotates at an angular velocity
motor

and drives the planet gear (P1) at an angular ve-
locity
s
haft motor



where 1

 is the speed reduce-
tion of the transmission between the motor M1 and the
right half of the shaft (2). Thus power transmission is
performed through P1-S1 towards the mass ‘a’. Similarly,
the rest half of the power produced by the motor M1 is
transmitted through P1-S2 towards the other mass ‘b’. A
characteristic of this mechanism is that the second planet
gear, P2, is fixed thus causing rolling of the spin gears S1
and S2 on P2. Obviously, the rotation of the planet gear
P1 enforces the spin gear S1 to rotate about its local axis
(initially coinciding with the global z-axis) and also en-
forces the casing (11) to rotate around x-axis.
Figure 2. Sketch of the proposed mechanism, in which the
lumped masses ‘a’ and ‘b’ are driven by the electric motors
M1 and M2 through a motion transmission system.
Figure 3. Abstractive sketch of the planetary system. The mechanical power flows through the planet gear (P1) to the spin
gears (S1) and (S2). The lumped masses ‘a’ and ‘b’ are attached to the free ends of the rods (4) and (5), respectively, which
are firmly connected with the aforementioned spin gears (S1) and (S2).

C. G. PROVATIDIS 651
When assuming the same diameters m
R
of the four
gears (P1, P2, S1 and S2), due to the aforementioned
rolling at the interface between P2 and S1, the following
conditions occur:
 the spin gear S1 rotates at an angular velocity

, which is half of that of the shaft (2
shaft
 
2
motor


),
 the spin gear S2 rotates at the same angular velocity
but of opposite sign,

,
 the casing rotates at the same angular velocity,

.
2.3. Design Parameters
Referring again to Figure 3, the characteristic dimen-
sions of the mechanism are:
 the radius r of the level (rod length) where the ma-
sses are attached and
 the radius R of the casing; more accurately it should
be the distance between the centroids of the masses
‘a’ and ‘b’.
The position of the concentrated masses, ‘a’ and ‘b’,
are determined through the angular position t



of
the ‘a’-rod (No.4) with respect to the negative x-axis
(convention of the positively oriented angle is

(Ox,
Oy). The configuration in Figure 3 corresponds to the
initial time, t = 0. In this case, the coordinates of the
masses ‘a’ and ‘b’ are




0
,, ,0,
aaa
t
x
yzr R
 and




0
,, ,0,
bbb
t
x
yzrR

, respectively, corresponds to the
initial time, t = 0. In more details, during the time inter-
val ‘t’, not only the two rods rotate around the axes of the
spin gears but also the casing in such a way that
casing

 . In the general case dealt in this paper, the
entire system rotates about the z-axis at an angular veloc-
ity
z

, using a second motor M2 shown in Figure 1 as
well as in Figure 2.
3. Parametric Equations of the
Figure-Eight-Shaped Path
At any time instance, in order to reach the final position,
starting from the abovementioned position

0
,,
aaa
t
xyz



,0,rR , the mass ‘a’ undertakes three simultaneous
motions. The first motion is the rotation of the spin gear
S1 at an angle 1t

  about z-axis, the second one is a
rotation of the casing around x–axis at an angle 21



,
while the third one is a rotation around the vertical
z-axis at an angle 3
z
t

 . As a result, when perform-
ing all three aforementioned rotations, at any time in-
stance t the resultant rotation matrix becomes:
23 1201aRRRR (1)
where
01
cossin 0
sincos 0
001
tt
tt














R (2)
12
10 0
0cos sin
0sin cos
tt
tt
















R (3)
23
cossin 0
sin cos 0
001
zz
zz
tt
tt














R (4)
Therefore, the coordinates of the mass ‘a’ are analyti-
cally given by:






0
2
cos cossincossinsin
cos sinsin cossincos
sin cos
aa
aaa
aa
t
zz
zz
xt x
yt y
zt z
rt trttRtt
rt trttRtt
rtRt
 
 



 
 
















R
(5)
Similarly, the coordinates of the mass ‘b’ are analyti-
cally given by:







0
2
coscossin cossinsin
cos sinsincossincos
sin cos
bb
bbb
bb
t
zz
zz
xt x
yt y
ztz
rt trttRtt
rt trttRtt
rtRt
 
 




 
















R
(6)
Details about the figure-eight-shaped curve are pro-
vided in Appendix A.
4. Velocities and Accelerations
Differentiating (5) in time, the velocity components of
the mass particle ‘a’ become:




sincoscos sin
cos 2cossin
cossincos
azzz
z
zz
x
trttrt t
rtRtt
rtR t t

 
 
 




(7)




sin sin-cos cos
cos 2coscos
cossin sin
azzz
z
zz
y
trt trtt
rtRt t
rtRt t

 
 
 




(8)




sin2 sin
a
z
trtRt

  
, (9)
Further differentiating in time, the acceleration com-
ponents of the mass ‘a’ become:

C. G. PROVATIDIS 652



 

22
z
22 22
cos cos
4cos2
sinsin+2cos2+ coscos
az
zz z
zz z
xtr tt
R
rt r
ttrtRt t
 
 
 

 




(10)






22
2
2
+cossin+2sincos
2sin2 tsincos
+2cos2+ cossin
+cos+sintcos
az zzz
z
zz
zz
ytrttrtt
rRtt
rtRt t
rtRt



 
 



(11)
 
22cos2cos
a
ztrtR t

  
 (12)
Differentiating (6) in time, the velocity components of
the mass ‘b’ are given by:




sincoscos sin
cos 2cossin
cossin cos
bzzz
z
zz
x
trttrtt
rtRtt
rtR t t

 

 
 
 
 

(13)



sin sincos cos
cos 2coscos
cossin sin
bzzz
z
zz
y
trttrtt
rtRt t
rtRtt

 

 
 
 
 

(14)




sin2 sin
b
ztrtRt

  
 (15)
By further differentiation in time, the acceleration
components of the mass ‘b’ become:





2
2
2
2
cos cos2sin sin
cos cos
4 sincossinsin
2cos2coscos
sin cossinsin
bzzz
zz
z
zz
zz
x
trttrtt
rt t
rt tRtt
rtRt t
rt tRtt

 


 


 


 
 

(16)






22
2
2
cos sin
2sincos
2sin 2sincos
2cos 2cossin
cossin cos
bzz
zz
z
z
z
zz
ytr tt
rt t
rtRt t
rtRtt
rtRt t
 
 




 





(17)
 
22cos2 +cos
b
ztrtR t

 
 (18)
5. Inertial Forces
Based on the abovementioned kinematics, the compo-
nents of the inertial force exerted on the mass ‘a’ can be
calculated by:
,,
xaa yaa zaa
FmxFmyFmz 
  (19)
while those on the mass ‘b’:
,,
xbb ybbzbb
F
mx Fmy Fmz     (20)
Substituting (10)-(12) into (19) and (16)-(18) into (20),
the resultant force components are given by:

22
4sinsin24coscos2
xxaxb
zz zz
FF F
ttt t
mr

 

 
 





(21)

22
4cossin24sincos2
yya yb
zz zz
FF Fmr
ttt t

 
 






(22)
2
4cos2
zza zb
F
FF mrt

  
 (23)
Equation (23) depicts that:
1) the z-component of the resultant inertial force is
proportional to the radius r, and
2) the maximum upward force
z
F
 (appearing at
0


) is equal and opposite to the maximum
downward (appearing at 12t

  ) value.
It can be also noticed that the geometrical parameter R
is not included in (21)-(23).
Remark: When considering a second couple of equal
masses ‘a'’ and ‘b'’ at a phase-difference 2

  , it is
trivial to verify that the corresponding forces
x
F


and
y
F


, which are given by (21) and (22) respectively,
ideally cancel those of the first couple (a,b).
6. Elastic Deformation of Rods
6.1. Axial Forces
In order to determine the axial component of the inertial
force vector


,,
T
kxkykzk
F
FF kabF, at a certain
mass, ‘a’ or ‘b’, it is necessary to consider the direction
cosines along the rod axes, which are given by:




/
/
/
rxaa a
ryaa a
rzaa a
nxxr
nyyr
nzzr






(24)
and




/
/
/
rxbb b
rybb b
rzbb b
nxxr
nyyr
nzzr






(25)
where
sinsin,sinsin
sin cos,sin cos
cos, cos
azb z
azbz
ab
x
Rt txRt t
yRttyRtt
zR tz Rt









 

(26)
denote the coordinates of those ends of the rods that do
not carry the concentrated masses.

C. G. PROVATIDIS 653
6.2. Rod Deformation (Quasi-Static Analysis)
Considering an instantaneous deformation of the elastic
rod ‘a’ according to Hooke’s law (semi-static assump-
tion), at every position the elongation of the rod that cor-
responds to the mass particle ‘a’ is given by:
ra
a
Fr
rEA  (27)
and therefore the updated variable radius is given by:
aa
rr r
 (28)
Similarly, for the rod that corresponds to the mass par-
ticle mass ‘b’ it holds:
rb
b
Fr
rEA  (29)
and therefore the updated variable radius is given by:
bb
rr r
 (30)
In order to simplify the subsequent analysis, without
loss of generality we assume that 0R. In this case, the
radial forces become:




222 23
cos1sin2cos
ra zz
F
mrt tt

   


(31)




222 23
cos1sin2cos
rb zz
F
mrt tt

    

(32)
Also, the semi-static deformations become:



2222 2
3
cos1 sin
2cos
az
z
m
rrt t
EA
t

 
 
 

(33)
and therefore the updated variable radius is given by:



23
22 2
22 422cos
cos 2
cos2 cos
a
z z
m
rr
EA
rt
rt
rt t  







 (34)
Similarly, for the rod that corresponds to the mass par-
ticle mass ‘b’ it holds:


2222 2
3
cos1sin
2cos
bz
z
m
rrt t
EA

 
 
 


 (35)
and therefore the updated variable radius is given by:



22 2
22422 3
cos 2
cos2 cos2cos
b
zz
m
EA
rr r t
rt trt






 (36)
Consequently, due to the change in the radii, z and
b
r, the updated vertical forces become:

2
2cos2
za aa
Fmztmr t



 
 (37)

2
2cos2
zb bb
Fmztmr t


 
 
 (38)

2
2cos2
zzazb ab
F
FFmrrt


 
  (39)
Substituting (34) and (36) into (39), one obtains:


 
22 22
22 42
2cos4cos2
cos2cos
z
z
rt
EA
m
Fm tr
rtt

 



 




(40)
Equation (40) can be split into two parts as follows:
,,
z
zrigid ztension
FF F


 (41)
with
2
,4cos2 zrigid
Fmrt


 (42)
representing the vertical resultant force due to the ri-
gid-body motion of the rods, and
 
22
,
22222 42
4cos2
2coscos2cos
ztension
z
m
Ft
EA
rtrtt








(43)
representing the contribution of the tension at the two
elastic rods, which carry the mass particles ‘a’ and ‘b’.
7. Impulse
The impulse caused by the vertical resultant force


z
F
t

is given by:
 
0
t
zz
ItF d



 (44)
Integrating (40) over time, after manipulations the
impulse (i.e., (44)) can be analytically expressed in clo-
sed form, using one term for the rigid-body part and an-
other for the tensile part, as follows:
,,
z
zrigid ztension
II I (45)
where
,2sin2 z rigid
Imr

  (46)
and
0246,sin 2sin 4sin 6 ztension ccc cI

  (47)
with


2
222
0
2
222
2
2
222
4
2
22
6
93
8
4
24
z
z
z
z
t
m
cr
EA
m
cr
EA
m
cr
EA
m
cr
EA














(48)
Obviously, the harmonics (sin2, sin 4,sin 6

 ) in (47)

C. G. PROVATIDIS 654
lead to zero values every 180, 90 and 60 degrees, respec-
tively, thus not practically contributing to the propulsion.
Concerning particularly the second harmonic (sin 2

),
not only the elastic part of the impulse vanishes every
180 degrees, but also that caused by the rigid-body mo-
tion (cf. Equation (46)). However, in addition to the three
harmonic terms, (47) includes also the term 
0
ct

,
which increases linearly with the time t. The analytical
expression of the term 0 c in (48) depicts that for a given
angular frequency

and given elasticity properties of
the two rods, the value of the net impulse is fully con-
trolled by the angular frequency z

.
8. A Preliminary Experimental Result
The prototype device weights approximately 22 kg in-
cluding all its structural members and the electric motors.
For reasons of functionality, the elastic bars (member No.
4 and member No. 5 in Figure 2) were manufactured
adequately thin, of 5.5 mm diameter and of 200 mm len-
gth, made of steel. Each lumped mass was taken appro-
ximately equal to 20 grams. The prototype device was
put in the center of the horizontal platform of an elec-
tronic scale, which was equipped by four identical strain-
gages at its four corners. The mean average of these four
sensors was shown on a digital display, with an accuracy
of ± 10 gr.
The experimental validation was a very difficult task,
mainly due to the high angular velocities required to ov-
ercome the gyroscopic phenomena appearing in the pro-
totype and the high bending occurring in the particular
choice of thin cylindrical bars. Nevertheless, even for the
abovementioned small masses, and even for the very
slow angular velocities (100

300 rpm) that were al-
lowed so as to avoid collision (of the members No.4 and
No.5 on the horizontal shaft No.2, shown in Figure 2),
the resultant impulse obtains a nonzero value. Clearly,
for a time-interval of 124 seconds in which 618 meas-
urements were automatically recorded as shown in Fig-
ure 4, the relative difference of the sum of the negative
values with respect to the sum of the positive values was
found about 11.6%, a fact that sustains the findings of
the abovementioned theoretical analysis.
9. Discussion
It is remarkable that the findings of this paper are in con-
sistency with previous experimental results (8% weight-
reduction) related to ideally rigid gyroscopes [16]. Since
the proposed device is essentially a flexible (elastic) gy-
roscope of which the masses operate in a hemisphere, it
is believed that similar accurate experiments with th-
ose of [16] should be performed for the current device.
Figure 4. A digital record of the ground reaction (in dN or
kg-force), for a time interval of approximately 2 minutes,
using two lumped masses each of 20 gr and low angular
velocities: approximately

= 100 rpm and
z

= 320
rpm. The amplitude of the ground force is close to 0.2
kg-force (i.e., 0.2 dN, equivalently, 2 N).
It is also believed that the proposed idea is a mechanical
alternative to older relativistic thoughts of 1960s, which
have recently revived [17].
In addition to the experimental shortcomings men-
tioned in Section 8, the weaknesses of the approximate
model presented in this study are as follows:
 Only the action of the concentrated masses has been
considered, while the moment of inertia of the rods
has been omitted.
 The axial deformation of the rods has been assumed
to coincide with that of static conditions immedi-
ately imposed at every time instance, while a more
accurate semi-analytical approach had to consider
dynamic response caused by longitudinal and trans-
verse traveling elastic waves along the rods.
 The influence of the bending deformation of the el-
astic bars has not been considered.
 The angular velocities

and z

have been con-
sidered to be constant while an accurate simulation
model would require the dynamic modeling of the
motors themselves or/and the use of power-to-an-
gular velocity curves [18,19].
 For the particular setup of this paper, at


2,3 2  the masses ‘a’ and ‘b’ are found at the
point of intersection I  0, 0,
R
r. This ‘collision’
could be theoretically avoided considering one mass
being of male and the other of female type. Obvi-
ously, in the general case where 0R this short-
coming is easily overcome but this selection leads
to more complicated analytical expressions that do
not offer further insight or any other essential ad-
vantages.
Concerning the mathematical analysis of this work, of
course a more accurate analysis has to be performed on

C. G. PROVATIDIS 655
the basis of a finite element elastodynamic model of the
entire structure in which not only tension but also bend-
ing and torsion will be automatically considered. A spe-
cial feature of such an analysis is the dependence of the
inertial forces on the radius, which is not a-prio ri known.
Furthermore, concerning the experimental part of this
work, it is worth-mentioning that not in all cases the ex-
periments were of the same quality as that presented in
Figure 4. In general, in order to achieve a remarkable net
impulse, a proper synchronization between the two an-
gular velocities,

and z

, is required. In order to in-
crease the magnitude of the net impulse, one could in-
crease either the angular velocities or the magnitude of
the lumped masses. In both cases the high bending in-
volved requires thick rods (high diameters) as well as
motors of high electric power. Therefore, due to the lim-
ited power capacity and, since the overall mechanical
strength of the experimental setup could not be improved
we were forced to stay with preliminary measurements
only. Of course, a future study should consider an en-
tirely different experimental setup.
Summarizing, the aforementioned advanced elastody-
namic model has also to be compared with high precision
experiments on a well-designed experimental device, a
fact that presupposes high technical skills and high-tech
premises, probably within an advanced industrial envi-
ronment.
10. Conclusions
This work contributes to the field of inertial propulsion,
proposing a new concept for the production of net im-
pulse through rotating masses. In the beginning, it was
shown that when a lumped mass moves along a circum-
ference, it repeats its position every 360 degrees, thus its
initial linear momentum is repeated and no net impulse is
finally produced for a whole revolution; this case corre-
sponds to the notorious ‘Dean drive’. Then, it was theor-
etically shown that when two lumped masses move along
a specific figure-eight-shaped path at a phase difference
of 180 degrees, in such a way that the latter path lays on
the surface of a hemisphere that additionally spins about
its axis of symmetry, the involved inertial forces lead to a
non-vanishing net impulse. Intuitively, this claim is true
because when the ratio of the spinning angular velocity
over the first one (formation of the figure-eight-shaped
path) is not an integer number, every mass does not re-
peat its initial position; in other words, when the mass
completes the figure-eight-shaped path (every 360 de-
grees), the linear momentum has a different value that
what it had at the initial position. In terms of combined
structural mechanics and kinematics, every mass is con-
nected to the center of the hemisphere through an elastic
rod that is imposed to highly variable tensile deformation.
This finding implies a variable radius of the hemisphere
and is decisive to produce net impulse towards the axis
of symmetry of the hemisphere. In addition to the theo-
retical findings, for purposes of demonstration and vali-
dation, a prototype mechanical device, capable of pro-
ducing the two aforementioned rotations, was manufac-
tured and fully described in this paper. Although the
need for more realistic experimental tests has been dis-
cussed, preliminary measurements are in consistency
with the proposed theory and sustain the production of
net impulse using rotating masses.
11. References
[1] H. Yoshikawa, T. Kagiwada, H. Harada and M. Mimura,
“Improvement of Propulsion Mechanism Based on the
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wards Synthesis of Micro-/Nano-Systems, Springer, Lon-
don, 2007, pp. 333-334.
[2] J. M. Gilbert, “Gyrobot: Control of Multiple Degree of
Freedom Underactuated Mechanisms Using a Gyrating
Link and Cyclic Braking,” IEEE Transactions on Robot-
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[3] P. R. Ouyang, Q. Li and W. J. Zhang, “Integrated Design
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[4] N. L. Dean, “System for Converting Rotary Motion into
Unidirectional Motion,” US Patent 2886976, 19 May
1959.
[5] I. I. Blekhman, “Synchronization in Science and Tech-
nology,” ASME Press, New York, 1988.
[6] I. I. Blekhman, P. S. Landa and M. G. Rozenblum,
“Synchronization and Chaotization in Interacting Dy-
namical Systems,” Applied Mechanics Reviews, Vol. 48,
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[7] I. I. Blekhman, A. L. Fradkov, H. Nijmeijer and A. Yu.
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[8] G. Y. Stepanov, “Why is it Impossible to Have ‘Dean’s
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[9] I. I. Blekhman, “Vibrational Mechanics: Nonlinear Dy-
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[10] M. G. Millis and N. E. Thomas, “Responding to Me-
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[11] M. G. Millis, “Assessing Potential Propulsion Break-
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C. G. PROVATIDIS 656
2010, pp. 34-41.
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[14] C. G. Provatidis and V. Th. Tsiriggakis, “A New Concept
and Design Aspects of an ‘Antigravity’ Propulsion
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C. G. PROVATIDIS 657
APPENDIX A
Remarks concerning the figure-eight-shaped curve
For the purpose of completeness, details are provided here for
the figure-eight-shaped curve.
First, in the absence of the spinning motion, i.e., when
0
z
 , a careful inspection of (5) and (6) reveals that:
1) At the initial time instance (0t), in fact the rods obtain
their horizontal position parallel to x-axis, i.e., ‘a’ on the
left and ‘b’ on the right side.
2) At the time instance given by 2t
 , both masses
obtain their vertical position.
3) At the time instance given by t

 , the masses mutu-
ally interchange their (horizontal) position.
4) At the time instance given by 32t  , the masses are
again found at their vertical position.
5) At the time instance given by 2t

 , the masses obtain
their initial (horizontal) position, and so on.
6) Therefore, the distance between the two masses varies
from the minimum 2
R
(vertical position) to the maxi-
mum value 22 2rR (horizontal position).
7) Both mass particles, ‘a’ and ‘b’, share a common path.
This happens because when putting ab tt t

 
in Equation (5), then it becomes identical with Equation
(6). In other words, the masses ‘a’ and ‘b’ move on the
same path and appear a constant phase difference of 180
degrees, thus they are mutually interchanged.
8) During the first 90 degrees (1 0a

 ), the mass ‘a’
draws the blue line while ‘b’ draws the red line of the
common path (Figure 5(a)). In the next 90 degrees the
situation is reversed, so as the mass ‘a’ follows the read
and the mass ‘b’ follows the blue line.
9) The common path intersects itself at the unique point I
 ,0,
R
r, which corresponds to 12
a

  or 1a 

32

, and so on.
10) All points of the abovementioned path belong to a
sphere of radius, i.e.:
222 2222
aa abbb sphere
x
yzxyz r , with 22
sphere
rrR
Second, in the case of a non-vanishing spinning angular ve-
locity (0
z


), the lumped masses do not generally follow the
same path, as clearly shown in Figures 5(b)-5(f), particularly
when the ratio z

 of the angular velocities is not an integer
number.
Figure 5. Perspective view of the paths drawn by the
lumped masses (r = 80 mm, R = 0), which are produced at
different synchronizations (ratio of angular velocities,
z


): (a) ratio = 0, (b) ratio = 0.5, (c) ratio = 1.0, (d) ratio
= 2.0, (e) ratio = 3.0 and (f) ratio = 3.5. In all cases, the blue
and red lines correspond to the masses ‘a’ and ‘b’ (shown in
Figure 2), respectively. The cases (a) and (d) are shown for
the first 180 degrees (0t



) so as to avoid overlap-
ping, while the rest cases for the first 720 degrees
(04t



 ).