*Relly Victoria Virgil Petrescu*

*IFToMM**, Romania*

*E-mail: rvvpetrescu@gmail.com*

*Florian Ion Tiberiu Petrescu*

*IFToMM**, Romania*

*E-mail: fitpetrescu@gmail.com*

*Submission: 11/10/2018*

*Revision: 11/21/2018*

*Accept: 11/28/2018*

*ABSTRACT*

Today,
robots are increasingly present in the machine building industry, sometimes
even in some sections to replace workers altogether, due to the high quality of
their work, repetitive, without stopping or pausing, without any manufacturing
and assembly scuffs. In this paper, one presents the mechanisms with bars and
gears, which are planetary mechanisms for robot automation and mechatronics,
structurally-topological. The gears and bars consist of at least one movable
articulated bar and one of the cylindrical, tapered or hipoidal
gears. Only gears with circular or straight toothed gears, in which the
relative position of the rotation or translation axes does not change, shall be
considered. The topological structure of the gears and gears is characterized
by a kinematic chain with articulated bars and at least one kinematic chain
with gears. The kinematic chain may be chain open (with a fixed rotation joint)
or closed chain (with at least two fixed joints). The kinematic chain with
gears is attached to the kinematic chain with bars so that at least two gear
wheels have centers in the bars of the bars and some wheels may be integral
with the bars. In practice, some of these gears with gears and gears are known
as planetary gears with cylindrical, conical or hipoidal
gears.

**Keywords**:
Robots; Mechatronic Systems; Structure; Topology

**1. ****INTRODUCTION**

Today, robots are increasingly present in the machine
building industry, sometimes even in some sections to replace workers
altogether, due to the high quality of their work, repetitive, without stopping
or pausing, without any manufacturing and assembly scuffs.

Additionally, robots do not get sick, do not require
medical leave or rest, work faster and better than humans and also support
toxic environments from dyers, general assemblies, etc.

Generally, robots have increased the
quality and productivity of work and have not even created a union to defend
their claims, demanding increased wages for them and larger holidays.
Interestingly, a robot is working without a break, but without salary, without
breaks, without complaining about working conditions in the plant.

Robots can work on three shifts,
that is, permanently, but not by shifting them like people did, but always
remaining the same robots deployed in operation, nonstop, for days, without
breaks, without rest, without problems.

It has come to the effect that the big car manufacturers
and even others, have entire sections in which only robots work. They do not
have to worry about each other, do not quarrel, do not complain, do not cry, do
not ask for the salary, do not require leave, they do not want free days and
can work with high returns and Saturday and Sunday, if necessary on three
shifts without a break.

The importance of implementing robots can no longer be
challenged. They have so increased the quality of work and the production of an enterprise that they can no longer give up
their help. Workers have reclassified themselves and work only in more friendly
workplaces, or in other workplaces, such as supermarkets, in better conditions,
with higher wages, with several days off and they are also pleased and all this
is due to production and additional gains from higher sales due to the robot
work in large factories.

We can clearly state that robots have improved our lives
considerably. Thanks to them, a new free day was introduced for almost all
working people, Friday, in addition to Saturday and we may soon be able to
introduce another free day, but we have to choose whether it is Monday or
Thursday.

People, in the beginning, were
taught by the trade union bosses to chase and sabotage the robots, to ruin them
and not to accept them. Today things are clear and the robots work quietly in
the big companies and factories for the sake of everyone, so now we can all
accept the silence of the automation, the robotics, the electronics, without
letting us be fooled by the union leaders, who slowly slow down and they will
calm down.

If we like it or not like, robots have already stolen all
our hard works places.

Anthropomorphic robots are, as I have already said, in
most of the most widespread and widely used works worldwide today, due to their
ability to adapt quickly to forced work, working without breaks or breaks 24 h
a day, without unpaid leave without asking for food, water, air, or salary.
Anthropomorphic robots are supple, elegant, easy to configure and adapted to
almost any required location, being the most flexible, more useful, more
penetrating, easy to deploy and maintain.

For the first time, these robots have asserted themselves
in the automotive industry and especially in the automotive industry, today
they have penetrated almost all industrial fields, being easily adaptable,
flexible, dynamic, resilient, cheaper than other models, occupying a volume
smaller but with a major working space. They can also work in toxic or
dangerous environments, so used in dyeing, chemical cleaners, in chemical or
nuclear environments, where they handle explosive objects, or in military
missions to land or sea mines, even if they were banned to use, because there
are still countries around the globe that use them, such as Afghanistan.

The most used today's industrial robots, is built. The
importance of the study of anthropomorphic robots has also been signaled, being
today the most widespread robots worldwide, due to its simple design,
construction, implementation, operation and maintenance. In addition,
anthromomorphic systems are simpler from a technological and cheaper point of
view, performing a continuous, demanding, repetitive work without any major
maintenance problems.

The basic module of these robots was also presented
geometrically, cinematically, of the forces, of its total static balancing and
of the forces that arise within or after balancing. In the present paper we
want to highlight the dynamics of the already statically balanced total module.
It has been presented in other works and studied matrix spatially, or more
simply in a plan, but in this case, it is necessary to move from the working
plane to the real space, or vice versa, passage that we will present in this
study.

In the basic plan module already presented in other
geometric and cinematic works, we want to highlight some dynamic features such
as static balancing, total balancing and determination of the strength of the
module after balancing. Through a total static balancing, balancing the
gravitational forces and moments generated by the forces of gravity is
achieved, balancing the forces of inertia and the moments (couples) generated
by the presence of inertial forces (not to be confused with the inertial
moments of the mechanism, which appear separately from the other forces, being
part of the inertial torsion of a mechanism and depending on both the inertial
masses of the mechanism and its angular accelerations.

Balancing the mechanism can be done through various
methods. Partial balancing is achieved almost in all cases where the actuators
(electric drive motors) are fitted with a mechanical reduction, a mechanical
transmission, a sprocket, spiral gear, spool screw type. This results in a
"forced" drive balancing from the transmission, which makes the operation
of the assembly to be correct but rigid and with mechanical shocks. Such
balancing is not possible when the actuators directly actuate the elements of
the kinematic chain without using mechanical reducers (ANTONESCU; PETRESCU,
1985; 1989; ANTONESCU *et al*., 1985a; 1985b; 1986; 1987; 1988; 1994;
1997; 2000a; 2000b; 2001; ATEFI et al., 2008; Avaei et al., 2008; AVERSA *et
al*., 2017a; 2017b; 2017c; 2017d; 2017e; 2016a; 2016b; 2016c; 2016d; 2016e;
2016f; 2016g; 2016h; 2016i; 2016j; 2016k; 2016l; 2016m; 2016n; 2016o; AZAGA;
OTHMAN, 2008; CAO *et al*., 2013; DONG *et al*., 2013; EL-TOUS, 2008;
COMANESCU, 2010; FRANKLIN, 1930; HE *et al*., 2013; JOLGAF et al., 2008;
KANNAPPAN et al., 2008; LEE, 2013; LIN *et al*., 2013; LIU *et al*.,
2013; MEENA AND RITTIDECH, 2008; MEENA et al., 2008; MIRSAYAR *et al*.,
2017; NG et al., 2008; PADULA; PERDEREAU; PANNIRSELVAM, 2008; 2013; PERUMAAL;
JAWAHAR, 2013; PETRESCU, 2011; 2015a; 2015b; PETRESCU; PETRESCU, 1995a; 1995b;
1997a; 1997b; 1997c; 2000a; 2000b; 2002a; 2002b; 2003; 2005a; 2005b; 2005c;
2005d; 2005e; 2011a; 2011b; 2012a; 2012b; 2013a; 2013b; 2016a; 2016b; 2016c;
PETRESCU *et al*., 2009; 2016; 2017a; 2017b; 2017c; 2017d; 2017e; 2017f;
2017g; 2017h; 2017i; 2017j; 2017k; 2017l; 2017m; 2017n; 2017o; 2017p; 2017q;
2017r; 2017s; 2017t; 2017u; 2017v; 2017w; 2017x; 2017y; 2017z; 2017aa; 2017ab;
2017ac; 2017ad; 2017ae; 2018a; 2018b; 2018c; 2018d; 2018e; 2018f; 2018g; 2018h;
2018i; 2018j; 2018k; 2018l; 2018m; 2018n; POURMAHMOUD, 2008; RAJASEKARAN et
al., 2008; SHOJAEEFARD et al., 2008; TAHER et al., 2008; TAVALLAEI; TOUSI,
2008; THEANSUWAN; TRIRATANASIRICHAI,
2008; ZAHEDI et al., 2008; ZULKIFLI et al., 2008).

In this paper one presents the space mechanisms with bars
and gears, which are planetary mechanisms for robot automation and mechatronics,
structurally-topological.

It is first considered the space mechanisms with the
kernel chain with open bars.

Two groups of such spatial mechanisms are known:
elementary mechanisms (with a single articulated bar) and complex articulated
mechanisms (with two or more articulated bars).

Elementary space mechanisms can be made with a single
central wheel (Figure 1) or two central wheels (Figure 2) whose fixed axes
coincide with the axis of the fixed hinge of the bar.

**2. ****METHODS AND MATERIALS**

It is first considered the space mechanisms with the
kernel chain with open bars.

Two groups of such spatial mechanisms are known:
elementary mechanisms (with a single articulated bar) and complex articulated
mechanisms (with two or more articulated bars).

Elementary space mechanisms can be made with a single
central wheel (Figure 1) or two central wheels (Figure 2) whose fixed axes
coincide with the axis of the fixed hinge of the bar.

The toothed wheels used in the space mechanisms are
conical wheels (Figure 1a) and worm wheel with screw and worm wheel (Figure
1b).

In the spherical space mechanism (Figure 1a), the central
conical wheel 1 engages the satellite cone wheel 2, their axes being competing
at the point S, this being the common tip of the rolling cones. The bar 3 has
two joints, one fixed in A0 (common to wheel 1), and another movable in A
through which it links to wheel 2.

If the axes of the two conical wheels are perpendicular,
the conical gear is called orthogonally, in this form being most often used in
practice.

The mobility of the spherical mechanism is M = 2, which
is deduced by the calculation of the custom formula (1):

(1)

a) b)

Figure 1: Elementary
space mechanisms can be made with a single central wheel

The range of space associated with this kinematic contour
is r = 3, since the axes of the rotation (m = 1) and roto-translation (m = 2)
are competing at point S.

Such a space mechanism with
a bar and a conical gear is equivalent to a spherical spherical joint with
monomobile joints, in which all axes are competing in the center S of the
sphere.

In the case of the worm gear (Figure 1b), the worm wheel
1 is a central wheel and forms a hypoid gear (worm) with the worm wheel 2, the
axes of the two gears being crossed in the orthogonal position. The bar 3 has a
fixed axle (denoted by A_{0}) common to that of the worm wheel 1, and
the mobile shaft of the joint A (with the worm wheel) is orthogonal to the
fixed one.

The mobility of the cross-axle space mechanism is M = 2,
resulting from the application of the formula (2) customized to the 6:

(2)

In the application of the formula (2) it is mentioned
that the engagement of the two worm wheels (1, 2) forms a pentamobile kemematic
coupling (m = 5), at which the contact of the two surfaces is made at a point.
A mechanism that includes a pentamobile kinematic coupling (maximum class) is
associated with the maximum gradient (r = 6).

The space mechanism equivalent to this worm gear
mechanism is an orthogonal spatial quadrilateral, the links of which are two
trimmed spherical couplings and two monoblock rotation couplings.

The elementary space mechanisms with two central conical
wheels (Figure 2) are obtained from the previous one (Figure 1a) by the
addition of a conical gear wheel 4, the axis of which is common to the fixed
one.

a) b)

Figure 2: Elementary
space mechanisms made with two central wheels

The first space mechanism (Figure 2a) contains a bar 3
and two conical gears (1, 2) and (2', 4) mounted in parallel. The mobility of
the mechanism is M = 2, this being calculated by the formula (3) for the
particular case of the spherical mechanisms:

(3)

The second spatial mechanism (Figure 2b) is a particular
case of the first mechanism from which it is obtained by orienting the movable
axis in a direction perpendicular to the fixed axis.

In this latter case, the gears 1 and 4 are equal and the
wheels 2 and 2' coincide, so that the two gears are mounted in series.

If the bar 3 is immobilized, the transmission ratio
between wheels 1 and 4 is obtained as the product of the partial transmission
reports to be written, in the general case (Figure 2a), according to the tooth
numbers in the form of:

(4)

For the particular case (fig. 2b), when and , of the formula (4) results , that is to say, the central wheels 1 and 4 are rotating in
the opposite direction, assuming that the bar 3 is immobilized.

The
rotation of the bar 3 is transmitted to the central wheels 1 and 4, so that the
formula:

(5)

we can deduce the
relationship:

(6)

By immobilizing one of the two central wheels 1 or 4, the
mobility of the spatial mechanism becomes M = 1. For example, if the wheel 4 is
immobilized, by actuating the rod 3 the movement is multiplied to the central
wheel 1, whose angular velocity is:

(7)

what is obtained from (6)
for .

In this case, the relative angular velocity of the wheel
2 relative to the bar 3 is deduced by writing the transmission ratio between
the wheels 2 and 4 under the immobilization of the bar 3:

(8)

from which it results .

a) b)

Figure 3: The space mechanism has at least two
articulated bars

The spatial gear mechanisms with conical gears are
obtained from the previously analyzed by the kinematic chaining operation with
bars.

By means of overlaying, the space mechanism has at least
two articulated bars (Figure 3), in which the conical gears are some (Figure
3a) or Orthogonal (Figure 3b).

The two kinematic schemes (Figure 3a, b) are isomorphic,
having the same topological structure, two bars (3 and 5) and three conical
gears (1, 2), (4, 5).

It is noted that the first two conical gears (1, 2) and
(4, 5) have the misaligned axes, being competing in the S1 point, and at the
third conical gear (2', 6) the axes intersect in S_{2}.

Also, the toothed wheel 5 is integral with the 5' bar
that makes the wheel hinge 6. The mobility of the two complex space mechanisms
is M = 3, a value resulting from the calculation using the custom formula (9):

(9)

For the numerical calculation of formula (9), for each of
the two kinematic schemes (figure 3), the following structural-topological
parameters were identified:

(10)

According to each mobility, there is a distinct kinematic
chain: the chain with bars (0, 3), the conical chain and bevel chain (0, 4,
5-5') and the conical toothed wheel chain (0, 1, 2-2', 6).

The three kinematic chains are linked together by the
common axes, a movable one for three elements (2, 3, 5) and another fixed for
four elements (0, 1, 3, 4).

It is found that the
three open kinematic outlines are partially coupled, so when the kinematic
chain (0, 1, 2-2', 6) is actuated, the other 2 chains are not driven in motion.

The action of the kinematic chain (0, 4, 5-5') influences
only the chain (0, 1, 2-2', 6), to which it imparts a first additional
movement.

By moving the kinematic chain (0, 3), the motion is
transmitted to the other two kinematic chains (0, 4, 5-5') and (0, 1, 2-2', 6)
of which the last chain receives a second movement further.

The calculation algorithm in the kinematic analysis of
this complex spatial mechanism (Figure 3), with mobility M = 3, highlights
three phases of work:

I) , when computed:

(11)

II) , when computed:

(12)

III), when computed:

(13)

**3. ****RESULTS AND DISCUSSION; SPATIAL MECHANISMS WITH BARS
AND CLOSED-CHAIN TOOTHED WHEELS**

This class of spatial mechanisms has, as a bar-shaped
chain, a articulated spherical quadrangular contour 4R, RCCR and RCCC spatial
quadrilateral, RRCCR spherical and spatial pentalater, RRRCRR spatial hexalator
and 7R spatial heptalater.

**3.1.
****Spherical
quadrangular space mechanisms. **

It is formed by overlapping the chain formed by two,
three and four conical gears.

Gears are distinct cinematic elements or are mounted in
solidarity with some bars of the spherical quadrangle contour.

It is considered the cranked spherical crank mechanism
(figure 4) to which a conical gear, two or three conical gears are attached.
The rocker bar 3 (BB_{0}) is perpendicular to the fixed rotation axis
projecting at point B_{0}.

a) b) c)

Figure 4: It is considered the crank-type spherical gear
mechanism to which is attached a conical gear, two or three conical gears

The variant 1 (figure 4a) is obtained by attaching the
orthogonal conical gear (2', 4) to the spherical (0, 1, 2, 3) quadrant so that
the wheel 2' is integral with the bar 2 and the wheel 4 has the axle fixed
joint with that of bar 3 with oscillating rocker movement.

The mobility of the spherical spatial mechanism is M = 1,
it is calculated by the formula (14) as the particular form:

(14)

where the numerical
values specific to the kinematic schemes (fig. 4a) were used (15):

(15)

The angular speed of the wheel 4 is calculated according
to the angular speeds of the bars 2 and 3 and the transmission ratio of the
conical gear (2', 4).

Variant 2 (Figure 4b) is obtained by attaching to the
articulated spherical quadrangle of the kinematic chain consisting of two
conical gears (1', 4) and (4', 5), in which the wheel 1' is integral with the
bar 1. The gear (1', 4') has the angle between the axes of wheels 1' and 4
equal to? (ABoB) formed by the axes of the joints of A and B. The gear (4', 5)
is also orthogonal. The mobility of the two-gear space mechanism is M = 1, the
value resulting from the calculation of formula (16) in the particular form:

(16)

in which the numerical
values of the structural-topological parameters were replaced (17):

(17)

Variant 3 (Figure 4c) consists of three conical gears in
which the wheel 4 is a distinct element with the common axis with the rod 1,
the wheels 5 (5') are freely mounted on the joint axis of A, the wheel 6 is
mounted free on the joint axis of B and the driven wheel 7 is freely mounted on
the fixed shaft of the articulation of B0. The mobility of this complex spatial
mechanism is M = 2, as it results from the numerical calculation using the
custom formula (18):

(18)

where the values specific
to the kinematic scheme were replaced (Figure 4c):

(19)

**3.2.
****Spatial
mechanisms with quadrature chain RCCR type**

The kinematic chain is formed with two articulations at
the base (A_{0}, B_{0}) and two cylindrical couplings (A, B)
with the orthogonal moving axes (Figure 5).

It starts from the RCCR space bar mechanism (Figure 5a)
which turns the crank 1 rotation into a limited rotation of the rocker rod 3.
The fixed axes of joints of A_{0} and B_{0} are perpendicular
to non-competing or competing.

The bar 2 consists of two orthogonal segments in S, each
having a direction parallel to the axis of one of the joints A_{0} and
B_{0}. These conditions determine the movement of the bar 2, which is a
circular translation in space. Since there is no rotation from the common
normal to the fixed axes of A_{0} and B_{0}, space associated
with the spatial kinematic outline (0, 1, 2, 3) is r = 5.

a)
b)

Figure 5: RCCR spacebar mechanism

The mobility of this mechanism is calculated by the
formula (20) customized in the form:

(20)

A kinematic chain with cylindrical gears (4, 5, 6', 7)
and conical (5', 6) are attached to this kinematic chain (0, 1, 2, 3) with
which they form the complex spatial mechanism with bars and gears (Figure 5b).

The mobility of the complex spatial mechanism is
calculated with the formula (21) written in the form:

(21)

It is to be noted that cylindrical gears 4 and 7 are
connected by cylindrical couplers to the shafts of the respective joints of A_{0}
and B_{0}.

The two mobilities are identified at bar 1 (as a crank)
and wheel 4 (as a rotation motion).

**3.3.
****Spatial
mechanisms with RCCC quadrilateral chain**

The RCCC quadrilateral cinematic chain space mechanism
(Figure 6a) has the axes arranged anyway in space, so that the additional chain
will comprise hippocidal gears (4, 5) and (5', 6) in which some hipod wheels
along the axis of rotation (Figure 6b).

a)
b)

Figure 6: The RCCC quadrilateral cinematic space
mechanism

**3.4.
****Spatial
heptalater chain mechanisms type 7R**

The kinematic chain with seven articulated bars forms a
heptagon closed contour and has the seven axes of rotation disposed anyway in
space (Figure 7a).

a) b)

Figure 7: Spatial
heptalater chain mechanisms type 7R

Each kinematic element is an articulated bar, the length
of which corresponds to the common normal at two neighboring rotation axes,
which are generally crossbars (nonconcurrent and nonparallel).

The spatial heptagon contour (A_{0}ABCDEE_{0}A_{0})
is materialized (Figure 7a) through the 13-sided spatial contour (A'_{0}A'AB'BC'CD'DE'EE'_{0}E0A'_{0}).

A closed cinema contour of seven bars, with the axes of
any joints, corresponds to an associated maximum height space (r = 6).

The range of the associated space is maximum (r = 6),
even if some of the seven axes are concurrent or parallel.

Such an articulated space mechanism (with all seven
kinematic couplings of class m = 1) is structurally topological equivalent to a
hypoid gear mechanism (with two couplings of class m = 1 and a coupling of
class m = 5, represented by punctual contact of the surfaces of the conjugate
teeth).

The mobility of the articulated space mechanism is M = 1,
which is verified by the calculation of the formula (22) customized as:

(22)

One or more splice chains with hypoid gears are attached
to this spatial kinematic chain (Figure 7b), the mechanism obtained is with
bars and hyboidal teeth (hyperboloidal).

In the case considered (Fig. 7b), three hypoid gears were
attached: the gear (7, 2 ') between the axles (D_{A}) and (D_{B}), the gear (8,9) between
the (D_{C}) 9 ', 10) between (D_{D}) and (D_{E}).

By attaching the three hypoid gears, three contours of
the highest rank are formed, so that the mobility of the complex space
mechanism with bars and hypoid gears is calculated with the formula:

(23)

In applying the above formula, it has been taken into
account that the wheels 7, 8, 9 (9 ') and 10 are freely mounted on respective
axes (D_{A}), (D_{C}), (D_{D}) and (D_{E}).

**3.5.
****Space
Spherical Mechanisms**

These spatial mechanisms are formed by attaching to a
spherical pentagonal chain two or more conical gears, obtaining several
variants with one, two or more mobility.

An example of a spherical cone with bevel gears and
conical gears with mobility one is shown below (Figure 8a).

The kinematic spherical chain with articulated bars
(Figure 8a) consists of movable elements 1, 2, 3 and 4 articulated between them
and connected to the fixed element 0 by the orthogonal axes of A_{0}
and C_{0}. A kinematic chain with wheels consisting of three conical
gears (5, 2'), (2', 6) and (6', 7) is attached to this kinematic chain.

The first two conical gears are represented in axial
projection, and the third conical gear appears in transverse projection (Figure
8a). The conical gear (6', 7) was also represented in axial projection (Figure
8b).

a) b)

Figure 8: Space
Spherical Mechanisms

The mobility of the complex spherical mechanism is
calculated using formula (24) in the particular form:

(24)

The two moves are represented by the independent rotation
of the input elements (bar 1 and wheel 5), and the driven element is the
toothed wheel 7.

**3.6.
****Spherical
Hexalathar Spacing Mechanisms**

Starting from the 5-ball spherical mechanism attached to
a kinematic chain with conical gears in several structural-topological
variants, of which there is a variant with four conical gears (Figure 9) with
the mobility M = 2.

The rods and gears consist of at
least one movable articulated bar and one of the cylindrical, tapered or
hipoidal gears.

It is first considered the space mechanisms with the
kernel chain with open bars.

Two groups of such spatial mechanisms are known: elementary
mechanisms (with a single articulated bar) and complex articulated mechanisms
(with two or more articulated bars).

Figure 9: Spherical Hexalathar
Spacing Mechanisms

Elementary space mechanisms can be made with a single
central wheel (Figure 1) or two central wheels (Figure 2) whose fixed axes
coincide with the axis of the fixed hinge of the bar.

The toothed wheels used in the space mechanisms are
conical wheels (Figure 1a) and worm wheel with screw and worm wheel (Figure
1b).

In the spherical space mechanism (Figure 1a), the central
conical wheel 1 engages the satellite cone wheel 2, their axes being competing
at the point S, this being the common tip of the rolling cones. The bar 3 has
two joints, one fixed in A0 (common to wheel 1), and another movable in A
through which it links to wheel 2.

If the axes of the two conical wheels are perpendicular,
the conical gear is called orthogonally, in this form being most often used in
practice.

The mobility of the spherical mechanism is M = 2, which
is deduced by the calculation of the custom formula (1):

The range of space associated with this kinematic contour
is r = 3, since the axes of the rotation (m = 1) and roto-translation (m = 2)
are competing at point S.

Such a space mechanism with a bar and a conical gear is
equivalent to a spherical spherical joint with monomobile joints, in which all
axes are competing in the center S of the sphere.

In the case of the worm gear (Figure 1b), the worm wheel
1 is a central wheel and forms a hypoid gear (worm) with the worm wheel 2, the
axes of the two gears being crossed in the orthogonal position. The bar 3 has a
fixed axle (denoted by A0) common to that of the worm wheel 1, and the mobile
shaft of the joint A (with the worm wheel) is orthogonal to the fixed one.

The mobility of the cross-axle space mechanism is M = 2,
resulting from the application of the formula (2) customized to the 6:

In the application of the formula (2) it is mentioned
that the engagement of the two worm wheels (1, 2) forms a pentamobile kemematic
coupling (m = 5), at which the contact of the two surfaces is made at a point.
A mechanism that includes a pentamobile kinematic coupling (maximum class) is
associated with the maximum gradient (r = 6).

The space mechanism equivalent to this worm gear
mechanism is an orthogonal spatial quadrilateral, the links of which are two
trimmed spherical couplings and two monoblock rotation couplings.

The elementary space mechanisms with two central conical
wheels (Figure 2) are obtained from the previous one (Figure 1a) by the
addition of a conical gear wheel 4, the axis of which is common to the fixed
one.

The first space mechanism (Figure 2a) contains a bar 3
and two conical gears (1, 2) and (2', 4) mounted in parallel. The mobility of
the mechanism is M = 2, this being calculated by the formula (3) for the
particular case of the spherical mechanisms.

The second spatial mechanism (Figure 2b) is a particular
case of the first mechanism from which it is obtained by orienting the movable
axis in a direction perpendicular to the fixed axis.

In this latter case, the gears 1 and 4 are equal and the
wheels 2 and 2' coincide, so that the two gears are mounted in series.

If the bar 3 is immobilized, the transmission ratio
between wheels 1 and 4 is obtained as the product of the partial transmission
reports to be written, in the general case (Figure 2a), according to the tooth
numbers in the form of.

By immobilizing one of the two central wheels 1 or 4, the
mobility of the spatial mechanism becomes M = 1. For example, if the wheel 4 is
immobilized, by actuating the rod 3 the movement is multiplied to the central
wheel 1, whose angular velocity is (7).

**4. ****CONCLUSIONS**

The rods and gears consist of at least
one movable articulated bar and one of the cylindrical, tapered or hipoidal
gears.

Only gears with circular or straight
toothed gears, in which the relative position of the rotation or translation
axes does not change, shall be considered.

The topological structure of the
gears and gears is characterized by a kinematic chain with articulated bars and
at least one kinematic chain with gears. The kinematic chain may be chain open
(with a fixed rotation joint) or closed chain (with at least two fixed joints).

The kinematic chain with gears is
attached to the kinematic chain with bars so that at least two gear wheels have
centers in the bars of the bars and some wheels may be integral with the bars.

In practice, some of these gears
with gears and gears are known as planetary gears with cylindrical, conical or
hipoidal gears.

The gearing of these gears in these complex mechanisms is
carried out in the form of series, parallel, or both parallel-series trains.

The
system is made according to the articulated plan cinematic chain, which can be
made as an open or closed kinematic chain.

**5. ****ACKNOWLEDGEMENT**

This text was acknowledged and appreciated by Dr. Veturia
CHIROIU Honorific member of Technical Sciences Academy of Romania (ASTR) PhD
supervisor in Mechanical Engineering.

**6. ****FUNDING INFORMATION**

a) 1-Research contract: 1-Research contract: Contract
number 36-5-4D/1986 from 24IV1985, beneficiary CNST RO (Romanian National
Center for Science and Technology) Improving dynamic mechanisms.

b) 2-Contract research integration. 19-91-3 from
29.03.1991; Beneficiary: MIS; TOPIC: Research on designing mechanisms with
bars, cams and gears, with application in industrial robots.

c) 3-Contract research. GR 69/10.05.2007: NURC in 2762;
theme 8: Dynamic analysis of mechanisms and manipulators with bars and gears.

d) 4-Labor contract, no. 35/22.01.2013, the UPB,
"Stand for reading performance parameters of kinematics and dynamic
mechanisms, using inductive and incremental encoders, to a Mitsubishi Mechatronic
System" "PN-II-IN-CI-2012-1-0389".

e) All these matters are copyrighted! Copyrights:
394-qodGnhhtej, from 17-02-2010 13:42:18; 463-vpstuCGsiy, from 20-03-2010
12:45:30; 631-sqfsgqvutm, from 24-05-2010 16:15:22; 933-CrDztEfqow, from
07-01-2011 13:37:52.

**7. ****ETHICS**

Authors should address any ethical issues that may arise
after the publication of this manuscript.

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PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.;
BUCINELL, R.; CORCHADO, J. (2017o) History of aviation-a short review. **J. Aircraft Spacecraft Technol.**, n. 1,
p. 30-49. DOI: 10.3844/jastsp.2017.30.49

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.;
BUCINELL, R.; CORCHADO, J. (2017p) Lockheed martin-a short review. **J. Aircraft Spacecraft Technol.**, n. 1,
p. 50-68. DOI: 10.3844/jastsp.2017.50.68

PETRESCU, R. V. V.; AVERSA,
R.; AKASH, B.; BUCINELL, R.; CORCHADO, J. (2017q)
Our universe. **J. Aircraft Spacecraft
Technol.**, n. 1, p. 69-79. DOI: 10.3844/jastsp.2017.69.79

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; CORCHADO, J.; Berto, f. (2017r) What is a UFO? **J. Aircraft Spacecraft Technol.**, n. 1, p. 80-90. DOI:
10.3844/jastsp.2017.80.90

PETRESCU, R. V. V.;
AVERSA, R.; AKASH, B.; CORCHADO,
J.; Berto, f. (2017s) About bell helicopter FCX-001 concept
aircraft-a short review. **J. Aircraft
Spacecraft Technol.**, n. 1, p. 91-96. DOI: 10.3844/jastsp.2017.91.96

PETRESCU, R. V. V.; AVERSA, R.; AKASH, B.; CORCHADO,
J.; Berto, f. (2017t) Home at airbus. **J. Aircraft Spacecraft Technol.**, n. 1, p. 97-118. DOI:
10.3844/jastsp.2017.97.118

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; CORCHADO, J.; Berto, f. (2017u) Airlander. **J. Aircraft Spacecraft Technol.**, n. 1, p. 119-148. DOI:
10.3844/jastsp.2017.119.148

PETRESCU, R.
V. V.; ERSA, R.; AKASH, B.; CORCHADO, J.; Berto, f. (2017v) When boeing is dreaming-a review. **J. Aircraft Spacecraft Technol.**, n. 1,
p. 149-161. DOI: 10.3844/jastsp.2017.149.161

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; CORCHADO, J.; Berto, f.( 2017w) About Northrop Grumman. **J. Aircraft Spacecraft Technol.**, n. 1,
p. 162-185. DOI: 10.3844/jastsp.2017.162.185

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; CORCHADO, J.; Berto, f. (2017x) Some special aircraft. **J. Aircraft Spacecraft Technol.**, n. 1, p. 186-203. DOI:
10.3844/jastsp.2017.186.203

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; CORCHADO, J.; Berto, f. (2017y) About helicopters. **J. Aircraft Spacecraft Technol.**, n. 1, p. 204-223. DOI:
10.3844/jastsp.2017.204.223

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; Berto, F.; Apicella, a. (2017z) The
modern flight. **J. Aircraft Spacecraft
Technol.**, n. 1, p. 224-233. DOI: 10.3844/jastsp.2017.224.233

PETRESCU, R.
V. V.; AVERSA, R.; AKASH, B.; Berto, F.; Apicella, a. (2017aa) Sustainable
energy for aerospace vessels. **J.
Aircraft Spacecraft Technol.**, n. 1, p. 234-240. DOI:
10.3844/jastsp.2017.234.240

PETRESCU, R. V. V.; AVERSA, R.; AKASH, B.;
Berto, F.; Apicella, a. (2017ab) Unmanned helicopters. **J. Aircraft Spacecraft Technol.**, n. 1,
p. 241-248. DOI: 10.3844/jastsp.2017.241.248

PETRESCU, R. V. V.; AVERSA, R.; AKASH, B.;
Berto, F.; Apicella, a. (2017ac) Project HARP. **J. Aircraft Spacecraft Technol.**, n. 1, p. 249-257. DOI:
10.3844/jastsp.2017.249.257

PETRESCU,
R. V. V.; AVERSA, R.; AKASH, B.; Berto, F.; Apicella, a. (2017ad) Presentation of Romanian engineers who
contributed to the development of global aeronautics-part I. **J. Aircraft Spacecraft Technol.**, n. 1,
p. 258-271. DOI: 10.3844/jastsp.2017.258.271

PETRESCU, R. V. V.; AVERSA, R.; AKASH, B.;
Berto, F.; Apicella, a. (2017ae) A first-class ticket to the planet
mars, please. **J. Aircraft Spacecraft
Technol.**, n. 1, p. 272-281. DOI: 10.3844/jastsp.2017.272.281

PETRESCU, R. V. V.; Aversa,
r.; Apicella, a.; Mirsayar, m. m.; Kozaitis, s. (2018a) NASA started a
propeller set on board voyager 1 after 37 years of break. **Am. J. Eng. Applied Sci.**, n. 11, p. 66-77. DOI:
10.3844/ajeassp.2018.66.77

PETRESCU, R. V. V.; Aversa,
r.; Apicella, a.; Mirsayar, m. m.; Kozaitis, s. (2018b) There is life on
mars? **Am. J. Eng. Applied Sci.**, n.
11, p. 78-91. DOI: 10.3844/ajeassp.2018.78.91

PETRESCU, R. V. V.; Aversa,
r.; Apicella, a.; PETRESCU, F. I. T. (2018c) Friendly environmental transport. **Am. J. Eng. Applied Sci.**, n. 11, p.
154-165. DOI: 10.3844/ajeassp.2018.154.165

PETRESCU, R. V. V.; Aversa,
r.; AKASH, B.; ABU-LEBDEH, T. M.; t. m.; Apicella, a. (2018d) Buses running
on gas. **Am. J. Eng. Applied Sci.**, n.
11, p. 186-201. DOI: 10.3844/ajeassp.2018.186.201

PETRESCU, R. V. V.; Aversa,
r.; AKASH, B.; ABU-LEBDEH, T. M.; t. m.; Apicella, a. (2018e) Some aspects
of the structure of planar mechanisms. **Am.
J. Eng. Applied Sci.**, n. 11, p. 245-259. DOI: 10.3844/ajeassp.2018.245.259

PETRESCU, R. V. V.; Aversa,
r.; ABU-LEBDEH, T. M.; APICELLA, A.; PETRESCU, F. I. T. (2018f) The forces of a simple carrier manipulator. **Am. J. Eng. Applied Sci.**, n. 11, p.
260-272. DOI: 10.3844/ajeassp.2018.260.272

PETRESCU, R. V. V.; Aversa,
r.; Abu-Lebdeh, T. m.; APICELLA, A.; PETRESCU, F. I. T. (2018g) The dynamics
of the otto engine. **Am. J. Eng. Applied
Sci.**, n. 11, p. 273-287. DOI: 10.3844/ajeassp.2018.273.287

PETRESCU, R. V. V.; Aversa,
r.; ABU-LEBDEH, T. M.; APICELLA, A.; PETRESCU, F. I. T. (2018h) NASA
satellites help us to quickly detect forest fires. **Am. J. Eng. Applied Sci.**, n. 11, p. 288-296. DOI:
10.3844/ajeassp.2018.288.296

PETRESCU, R. V. V.; Aversa,
r.; ABU-LEBDEH, T. M.; APICELLA, A.; PETRESCU, F. I. T. (2018i) Kinematics
of a mechanism with a triad. **Am. J. Eng.
Applied Sci.**, n. 11, p. 297-308. DOI: 10.3844/ajeassp.2018.297.308

PETRESCU, R. V. V.; Aversa,
r.; APICELLA, A.; PETRESCU, F. I. T. (2018j) Romanian engineering "on the
wings of the wind". **J. Aircraft
Spacecraft Technol.**, n. 2, p. 1-18. DOI: 10.3844/jastsp.2018.1.18

PETRESCU, R. V. V.; Aversa,
r.; APICELLA, A.; PETRESCU, F. I. T. (2018k) NASA Data used to discover eighth
planet circling distant star. **J.
Aircraft Spacecraft Technol.**, n. 2, p. 19-30. DOI:
10.3844/jastsp.2018.19.30

PETRESCU, R. V. V.; AVERSA,
R.; APICELLA, A.; PETRESCU, F. I. T. (2018l) NASA has found the most distant
black hole. **J. Aircraft Spacecraft
Technol**., n. 2, p. 31-39. DOI: 10.3844/jastsp.2018.31.39

PETRESCU, R. V. V.; AVERSA,
R.; APICELLA, A.; PETRESCU, F. I. T. (2018m) Nasa selects concepts for a new
mission to titan, the moon of saturn. **J.
Aircraft Spacecraft Technol.**, 2: 40-52. DOI: 10.3844/jastsp.2018.40.52

PETRESCU, R. V. V.; AVERSA,
R.; APICELLA, A.; PETRESCU, F. I. T. (2018n) NASA sees first in 2018 the direct
proof of ozone hole recovery. **J.
Aircraft Spacecraft Technol.**, n. 2, p. 53-64. DOI:
10.3844/jastsp.2018.53.64

Pourmahmoud, N. (2008) Rarefied Gas Flow Modeling inside Rotating
Circular Cylinder, **Am. J. Eng. Applied
Sci.**, v. 1, n. 1, p. 62-65. DOI: 10.3844/ajeassp.2008.62.65

Rajasekaran, a.; Raghunath,
p. n.; Suguna, K. (2008) Effect of Confinement on the Axial Performance of
Fibre Reinforced Polymer Wrapped RC Column, **Am. J. Eng. Applied Sci.**, v. 1, n. 2, p. 110-117. DOI:
10.3844/ajeassp.2008.110.117

Shojaeefard, m. h.;
Goudarzi, k.; Noorpoor, a. r.; Fazelpour, M. (2008) A Study of Thermal Contact using
Nonlinear System Identification Models, **Am.
J. Eng. Applied Sci.,** v. 1, n. 1, p. 16-23. DOI: 10.3844/ajeassp.2008.16.23

Taher, s. a.; Hematti, r.; Nemati,
M. (2008) Comparison of Different Control Strategies in GA-Based Optimized UPFC
Controller in Electric Power Systems, **Am.
J. Eng. Applied Sci.**, v. 1, n. 1, p. 45-52. DOI: 10.3844/ajeassp.2008.45.52

Tavallaei, m. a.; Tousi, B. (2008) Closed Form Solution to an Optimal Control
Problem by Orthogonal Polynomial Expansion, **Am. J. Eng. Applied Sci.**, v. 1, n. 2, p. 104-109. DOI:
10.3844/ajeassp.2008.104.109

Theansuwan, w.; Triratanasirichai,
K. (2008) Air Blast Freezing of Lime Juice: Effect of Processing Parameters, **Am. J. Eng. Applied Sci.**, v. 1, n. 1,
p. 33-39. DOI: 10.3844/ajeassp.2008.33.39

Zahedi, s. a.; Vaezi, m.; Tolou, N. (2008) Nonlinear
Whitham-Broer-Kaup Wave Equation in an Analytical Solution, **Am. J. Eng. Applied Sci.**, v. 1, n. 2,
p. 161-167. DOI: 10.3844/ajeassp.2008.161.167

Zulkifli, R.; Sopian, k.;
Abdullah, s.; Takriff,
M. S. (2008) Effect of Pulsating Circular Hot Air Jet Frequencies on Local and
Average Nusselt Number, **Am. J. Eng.
Applied Sci.**, v. 1, n. 1, p. 57-61. DOI: 10.3844/ajeassp.2008.57.61

**SOURCE
OF FIGURES:**

Petrescu and Petrescu, 2011b.