1. INTRODUCTION
The
kinematic scheme of a cardan transmission can be seen in figure (1). It is
composed of two cardan shafts (one input and one output), both of which are equipped
with a cardan cross (universal joint or universal joint). Between the two
universal couplings, a further (additional) cardan shaft (axle) is mounted.
This mechanism is designed to transmit the mechanical movement (within a
vehicle) from one bridge to the other. If the vehicle's motor is on the front
and with on the rear axle transmission, or vice versa when the vehicle's engine
is on the rear and the transmission is on the front axle, or when we have
multiple (multi-axle) transmission on heavy vehicles or 4x4 cars (PETRESCU,
2012b).
Figure
1: The cardan transmission
We
most often encounter this transmission on buses, trucks, trolleybuses, trains,
heavy cars, trucks, and all-wheel drive cars. It usually transmits the rotation
movement from the front axle to the rear axle. It is necessary because there
are big games in the transmission both left-right and up and down. It is the
only mechanism that can transmit a rotation motion on a long axis that moves at
the same time laterally and up and down. Generally, the yield of such a
transmission is quite high, even if the rotation speed varies within the
mechanism, but it is reconstituted at the end.
In
2010, more than 800 million vehicles circulate across the planet (ANTONESCU,
2000; ANTONESCU; PETRESCU, 1985; ANTONESCU; PETRESCU, 1989; ANTONESCU et al.,
1985a; ANTONESCU et al., 1985b; ANTONESCU et al., 1986; ANTONESCU et al.; ANTONESCU
et al., 1987; ANTONESCU et al., 1988; ANTONESCU et al., 1994; ANTONESCU et al.,
1997; ANTONESCU et al.; ANTONESCU et al., 2000a; ANTONESCU et al.2000b; ANTONESCU
et al., 2001; AVERSA et al.; AVERSA et al., 2017a; AVERSA et al., 2017b; AVERSA
et al., 2017c; ; AVERSA et al., 2017d; AVERSA et al., 2017e; MIRSAYAR et al.,
2017; PETRESCU et al., 2017a; PETRESCU et al., 2017b; PETRESCU et al., 2017c;
PETRESCU et al., 2017d; PETRESCU et al., 2017e; PETRESCU et al., 2017f;
PETRESCU et al., 2017g; PETRESCU et al., 2017h; PETRESCU et al., 2017i, 2015;
PETRESCU; PETRESCU, 2016; PETRESCU; PETRESCU, 2014; PETRESCU; PETRESCU, 2013a; PETRESCU;
PETRESCU, 2013b; PETRESCU; PETRESCU, 2013c; PETRESCU; PETRESCU, 2013d; PETRESCU;
PETRESCU, 2011; PETRESCU; PETRESCU, 2005a; PETRESCU; PETRESCU, 2005b; PETRESCU,
2015a; PETRESCU, 2015b; PETRESCU, 2012a; PETRESCU, 2012b; HAIN, 1971; GIORDANA
et al., 1979; ANGELES; LOPEZ-CAJUN, 1988; TARAZA et al., 2001; WIEDERRICH;
ROTH, 1974; FAWCETT; FAWCETT, 1974; JONES; REEVE, 1974; TESAR; MATTHEW, 1974;
SAVA, 1970; KOSTER, 1974).
2. UNIVERSAL JOINT (CARDAN COUPLING)
The
cardan coupling mechanism is a spherical mechanism (see figure 2) due to the
spherical motion imposed by any universal coupler. In a cardan coupling the
four rotation axes are competing in a point S (Figure 2); (PETRESCU, 2012b).
Figure 2:
Universal joint or cardan coupling
The
spherical mechanism is crank-type if the angle d1 is
the smallest in relation to the angles d2, d3, d0
(figure 2a). In the case of the cardan mechanism, the element 2 (element
representing a ball moving on a spherical surface with the center in S) is
materialized by the two moving axes D12 and D23 (figure 2b). The specificity of the cardan
mechanism is that the angles d1, d2, d3 are all 900 and the angle d0
between the two fixed axes is obtuse, the angle d=a (the attachment
of d0)
being sharpened (Figure 2c).
The
analytical calculation of the input-output transmission function is done by
means of the kinematic diagram of the cardanic mechanism (Figure 2c) and the
schematic diagram of the u1 and u2 versions of the axes D12 şi D23, where (figure 3a) is denoted by D1 axis
of the drive shaft and with the D2 axis of the driven shaft. With these notations
(Figure 3a), the transmission function between the input and output shaft
(driven) is of the form j2(j1) or y2(j1).
Figure 3:
Cardan transmission: Versors of the transmit function
For
this purpose one considers the mobile versor 
and 
(Fig. 3a), which are orthogonal and oriented
in the directions of the moving axes D12 and D32 competing in the center O = S (Figure 2).
The
versor rotates in the plane [y1z1]
and is positioned with the angle j1 from the y1 axis (Figure 3). The vector rotates with the angle j2=y2 in the plane [y2z2], being positioned
relative to the common axis z2 = z1 (Figure 3b).
The
two versors are analyzed analytically by their components on axes x1,
y1 and z1,2 (Figure 3a), according to the relations of
the system (1).
(1)
From
the 
(perpendicularity of the versors and )
condition, the relationship between their projections on the fixed axes (2) is
deduced.
(2)
Expression
(2) is written in the default form (3) in which the rotation angle of the
output shaft 2 is based on the angle of rotation of the input shaft 1, but also
according to the sharp angle a=d.
(3)
Then
the 0 transmission function gets the expression (4).
(4)
By
derivation the expression of reduced angular velocity (5) is obtained.
Initially the expression is also depending on 
, and at
the end it is expressed only according to 
(and of course by a).
(5)
By
a new derivation is obtained also the order of transmission 2, respectively the
reduced angular acceleration (relation 6).
(6)
The
extreme values of the reduced angular velocity of the driven shaft 2 are
obtained from its analytical expression with the limit conditions for the input
angle j1.
(7)
The
dual cardan mechanism is obtained by mounting two simple cardan shafts in
series, so that the two intermediate shaft forks are coplanar (see Figure 4).
Figure 4:
Cardan transmission (double cardan mechanism); the intermediate shaft is
observed
3. DOUBLE CARDAN MECHANISM
The
double cardan mechanism has the advantage of performing the synchronous
movement between the input shafts D1 and the output D3
(Figure 4a). The intermediate shaft with the axle D2 has
two fixed points 
and 
and so that it no
longer needs a material bonded to the fixed bed (Figure 4a).
For
the output shaft there are two positions (see Figure 4b): one 
, and another 
symmetrical with the axis 
in relation to the axis 
. The
synchronization of the two movements (input-output) can be proved by means of
the 0 transmission function which is written for the two simple cardan
couplings (figure 4a), (relation 8); (Petrescu, 2012b).
(8)
The
dual cardan coupling variant with the parallel axle output and input is
generally used on heavy goods vehicles (lorries) so that the distance between
the two axes may vary within certain limits (see Figure 5); In this situation a
variable length of the intermediate shaft 2 is required, constructively made by
a telescopic intermediate shaft.
Figure 5:
Double, synchronous and telescopic (with telescopic intermediate shaft)
The two
pieces of the axle 2, D2 şi D2’, have
the coplanar forks (Figure 5) and are connected by a transverse groove
coupling, allowing the variation of the length O102 and
the angle dimension a=d when the distance between the axes h13
varies.
If the two forks of the intermediate shaft D2 are
not coplanar (see figure 6), the movement of the driven shaft D3 is no
longer synchronous with that of the drive shaft D1.
Figure 6:
Double Asynchronous Cardan Transmission (intermediate shaft forks are not
coplanar)
4. CONCLUSIONS
Although
apparently the dynamic loading of the double (dual) transmission increases with
the mechanical (mechanical) inertial moment, the effect itself is negligible
under actual operating conditions (for normal cardan transmissions, built and
properly mounted).
The
elasticity of the intermediate shaft influences the homokinetic deviation of
the transmission as follows: a) in the usual cases the bicardan transmission
becomes quasi-homo-kinetic and therefore the deviation from homokinetic can be
virtually neglected; b) In special cases with long (or very long) intermediate
shafts and high (or very high) mechanical moments of inertia it is necessary to
offset the homokinetic deviation by designing the transmission so that the
deviation from homokinetic becomes null, or negligible. c) Under normal
operating conditions, the influence of the elasticity of the intermediate shaft
on torsion moments may be neglected.
5. ACKNOWLEDGEMENTS
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
Research
contract: Contract number 27.7.7/1987, beneficiary Central Institute of Machine
Construction from Romania (and Romanian National Center for Science and
Technology). All these matters are copyrighted. Copyrights: 394-qodGnhhtej
396-qkzAdFoDBc 951-cnBGhgsHGr 1375-tnzjHFAqGF.
7. AUTHORS’ CONTRIBUTION
All
the authors have contributed equally to carry out this work.
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