Enumeration, kinematic, static and mechanical efficiency analysis of differential screw mechanisms G. Ciccioli, E. Pennestr`ı, P.P. Valentini Dipartimento di Ingegneria Meccanica Universit`a di Roma tor Vergata Via del Politecnico, 1 00133 Roma, Italy e-mail:
[email protected] December 2008
Abstract Although differential screw mechanisms are widely used in precision mechanisms to control fine relative movements, their enumeration and systematic methods of analysis seem less widespread. In this paper methods to support the preliminary design steps of these devices are proposed. In particular, has been proposed an enumeration algorithm of screw mechanisms. This method makes use of kinematic structure relationships deduced by R. Kraus. Kinematic, static analyses of differential screw mechanisms are described mainly in the German technical literature. However, such analyses are based on graphical constructions. In the paper graph-based analytical methods of kinematic, static and mechanical efficiency analysis are proposed. Examples of application of the suggested methods are offered.
1
Introduction
This investigation addresses the enumeration of differential screw mechanisms and analysis of their kinematics, statics and mechanical efficiency. The simplest differential screw mechanism is composed of one revolute, two prismatic and two screw pairs with different pitch. Thus the slider displacement is proportional to the pitch difference. This type of devices is often used in precision measurement equipment, optical instruments and in all applications where micromovements are required. The paper has been divided in three main parts: 1. Deduction of analytical expression useful for the kinematic structure analysis of mechanisms. 2. Enumeration of differential screw mechanisms. 3. Formulation of equations to express the kinematics, the statics and mechanical efficiency.
1
A new method to enumerate and classify these mechanisms is proposed. Some novel kinematic chains have been discovered. The method, based on some noteworthy relations deduced by R. Kraus [4, 5], makes use of the polynomial characteristic isomorphism test of Mruthyunjaya and Raghavan [9] adapted to labeled graphs. In particular, the enumeration technique of R. Kraus has been extended to kinematic chains with reduced mobility parameter λ. Usually, λ = 6 and λ = 3 for spatial and planar kinematic chains, respectively. The enumeration technique of Kraus, originally developed for hand calculation, can be also adapted to automatic computation. Previous work in the field of enumeration of screw mechanisms is due to F. Reuleaux [1], W. Jahr [2] and P.W. Jensen [3]. The available methods of kinematic and static analysis of screw mechanisms are adressed mainly in German literature (e.g. [4, 5, 6]). These methods are of graphical nature. The kinematic analysis of differential screw mechanisms is also discussed in a more recent bibliographical source by P.W. Jensen (e.g. [3]), R. Beyer in his textbook [6] cites two papers [7, 8] he authored on the computation of mechanical efficiency. However, the authors did not have access to these papers. The methods for kinematic, static and mechanical efficiency analysis herein proposed are very systematic in nature and do not require graphical representation.
2
Brief historical overview on screw mechanisms
Figure 1: Kinematic structure of a Roman epoch screw press for squeezing grape berries Often screws are installed in press mechanisms to obtain high mechanical advantage. In Figure 1 is shown the kinematic structure of a screw press used in Roman epoch to press grapes. Using the screw as a basic component of complex mechanisms also attracted the attention of Leonardo da Vinci. In his notebooks there are several sketches (e.g. Figure 2) that witness his scientific interest in this device. 2
Figure 2: Differential screw mechanisms from Leonardo da Vinci (Madrid Codex) Figure 3 demonstrates that Leonardo was fully aware that the screw is a modification of the inclined plane and may be regarded as a right-angled triangle wrapped round a cylinder. The sketches describe different experimental setup prepared by Leonardo for computing the relationship between the torque on the screw and axial resisting force.
Figure 3: Experimental setup for testing the statics of the screw (Leonardo da Vinci, Madrid Codex) The range of applications of screws has been expanded in modern times to include devices delivering high precision movements in which the level of precision is of the order of a micrometer. In fact, differential screws provide a cheap and simple way to adjust precision movements of optical equipment.
3
Review of Kraus’ kinematic structure analysis formulas
The enumeration of mechanisms requires the availability of equalities and inequalities between the number of links, the number of kinematic pairs, the degrees-of-freedom and other relevant kinematic structure parameters. In this and the following section some relationships deduced by R. Kraus [4, 5] will be reviewed. These are indispensible to the proposed enumeration procedure of differential screw mechanisms.
3
The Kutzbach’s formula for the computation of the degree-of-freedom (dof) v of spatial mechanisms can be written as J X (λ − fi ) (1) v = λ (n − 1) − i=1
where - n is the total number of links; - fi is the dof of the generic kinematic pair; - λ is the mobility parameter, equal to 6 and 3 for spatial and planar and spherical mechanisms, respectively; - J the total number of kinematic pairs. Links and kinematic pairs can be broadly divided into two categories: a) adjacent to the frame link; b) intermediate or non adjacent to the frame link. A link is considered adjacent to the link frame when it shares a kinematic pair with such a link. Similarly, a kinematic pair is adjacent to the frame link if one of its kinematic elements is the frame link. Therefore, the following equality holds: n = 1 + w + n0 ,
(2)
where - w is the number of adjacent links and adjacent kinematic pairs; - n0 is the number of intermediate links. If we denote by - fw the dof of the generic adjacent kinematic pair; - f0 the dof of the generic intermediate kinematic pair; then
J X i=1
fi =
X
fw +
X
f0 .
(3)
For (λ = 6), the following equality holds J X i=1
(6 − fi ) = 6w −
X
fw + 6s0 −
where s0 is the number of intermediate kinematic pairs.
4
X
f0 ,
(4)
Therefore, from (1), taking into account (2) and (4), we have X X v= fw + f0 − 6 (s0 − n0 ) .
(5)
The total number Lind of independent circuits in a mechanism is Lind = J − n + 1
(6)
Hence, the number z of independent circuits where the link frame is absent is z = s0 − (n0 + w) + 1 . Substituting (7) into (5) follows X
f0 = 6 (z + w − 1) + v −
(7)
X
fw .
The number of kinematic elements of intermediate kinematic pairs is X X 2s0 = xnx + (x − 1) wx ,
(8)
(9)
where - nx is the number of intermediate links with multiplicity equal to x; - wx is the number of adjacent links with multiplicity equal to x.
3.1 Limitations on the number of kinematic pairs, links and independent circuits Purpose of this subsection is the derivation of algebraic relationships between the number of kinematic pairs, the number of independent circuits and the number of links. These expressions will be useful in the enumeration phase. Combining (7) with (9) and taking into account X X n0 = nx = n2 + n3 + nx ,
(10)
x≥4
one obtains 2
X
nx + 2z + 2
X
wx − 2 =
X
xnx +
X
(x − 1) wx .
Therefore we have X
(x − 2) nx +
X
(x − 3) wx − 2z + 2 = 0 .
Since X
(x − 2) nx = (2 − 2)n2 + (3 − 2)n3 +
X
(x − 2) nx ,
x≥4
the following relationship n3 = 2z − 2 −
X
(x − 2) nx −
x≥4
X
(x − 3) wx ,
for computing the number of intermediate ternary links is justified. 5
(11)
Similarly, substituting n3 into (10), follows n2 = n0 + 2 (1 − z) +
X
(x − 3) nx −
x≥4
X
(x − 3) wx .
This last equation, considering (7), can be rewritten as X X n2 = s0 + 3 (1 − z) + (x − 3) nx + (x − 4) wx .
(12)
(13)
x≥4
Since n2 ≥ 0, the following inequality must hold for the number of intermediate kinematic pairs X X s0 ≥ 3 (z − 1) − (x − 3) nx − (x − 4) wx . (14) x≥4
Moreover, considering equation (11) one obtains z =1+
n3 1 X 1X (x − 2) nx + (x − 3) wx . + 2 2 2
(15)
x≥4
Therefore, the minimum value of z is z ≥1+
1X (x − 3) wx . 2
Finally, since n3 ≥ 0, the following equality must hold X X (x − 2) nx ≤ 2 (z − 1) − (x − 3) wx .
(16)
(17)
x≥4
4
Kinematic structure analysis of planar mechanisms
The previous equations can be written for λ = 3 (planar mechanisms). The Kutzbach’s formula can be simplified as J X (3 − fi ) = 3 (n − 1) − v , i=1
or v = 3 (n − 1) −
J X
(3 − fi ) .
(18)
i=1
After setting f = 1 and
P
(3 − f ) = 2s, from (18) it follows the Gr¨ubler’s formula v = 3 (n − 1) − 2s ,
(19)
with s the total number of lower kinematic pairs. However, since s = w + s0 ,
(20)
v = w + 3n0 − 2s0 .
(21)
from (19) one obtains
6
For planar mechanisms the following qualities hold X f0 = s0 , X fw = w ,
(22) (23)
therefore, from (8) follows s0 = 3 (z − 1) + 2w + v ,
(24)
n0 = 2 (z − 1) + w + v .
(25)
or, taking (7) into account, Considering this last relationship, (12) can be simplified in the form X X n2 = v + (x − 3) nx + (x − 2) wx .
(26)
x≥4
5
Kinematic structure analysis of mechanisms with parameter mobility λ=2
There is a class of mechanisms whose mobility parameter λ can be set equal to two. For instance, all planar mechanisms with only prismatic pairs belong to this class. To this class belong all the screw mechanisms with all screw pairs having coincident axes. The kinematic structure formulas deduced in the previous sections will be first specialised for such a class of mechanims. Afterward these mechanisms will be enumerated. For λ=2, the Kutzbach’s formula takes the form v = 2 (n − 1) −
J X
(2 − fi ) .
i=1
Alternately, denoted by s the total number of kinematic pairs, one can write v = 2 (n − 1) − s .
(27)
n = 1 + w + n0 ,
(28)
s = w + s0 ,
(29)
Moreover, after we let
equation (27) gives v = w + 2n0 − s0 .
(30)
Moreover, after letting w=
X
fw ,
s0 =
X
f0 ,
from equation X
f0 = 2 (z + w − 1) + v − 7
X
fw ,
(31)
equivalent to the (8) valid for spatial mechanisms, one obtains s0 = 2 (z − 1) + w + v ,
(32)
n0 = z + v − 1 .
(33)
and Equations (11) and (12) do not change since in their derivation the parameter λ is not involved. However, once (33) is considered, (12) can be rewritten in the form X X n2 = 1 − z + v + (x − 3) nx − (x − 3) wx .
(34)
x≥4
Similarly, the inequalities (16) and (17) are not affected by the reduced value of λ.
6
Enumeration of kinematic chains with screw pairs
The previous formulas can be used for the enumeration of kinematic chains for any given value of the parameter λ. In our case, λ is set equal to two. This reduced value of λ is characteristic of: • mechanisms with prismatic pairs only; • mechanisms with revolute, prismatic and screw pairs. The present treatment differs from the one proposed by Kraus since the frame link, denoted by a vertex labeled with a, is not a-priori chosen. The data required for the enumeration are: • the number w of intermediate links and their multiplicity; • the degree-of-freedom v of the mechanism; • the number z of independent circuits where the frame link is absent. Case a) : w = 2w2 , the two adjacent links are both binary, v = 1, z ≥ 0. Case a1 z = 0, s0 =1, n0 =0 (see Figure 4a) Case a2 z = 1, nmax =1 n4 , s0 =3, n0 =1, n3 =2, n2 =-1. Does not exists. Case b) : w = 2w2 + 1w3 , two binary links and one ternary link, v = 1, z ≥ 0. Case b1 z = 0, s0 =2, n0 =0 (see Figure 4b) Case b2 z = 1, n2 =-1. Does not exists. Case c) : w = 3w2 , three adjacent binary links, v = 2, z ≥ 0. Case c1 z = 0, s0 =3, n0 =1, n3 =1, n2 =0 (see Figure 4c)
8
a a
a
A1
e
A2 A1
A2
A1
A6
b
A3
A5
b
A5 A2
A4
A4 b
A3
c
a)
d
c
d
A3
c
c)
b)
Figure 4: Graphs of mechanisms Enumeration steps are as follows: • the type of kinematic pairs is established; • the edges of the graph are labeled in all nonisomorphic ways; In this investigation these steps are computer programmed to guarantee the completeness of the results. The Figure 5 summarizes the steps of the enumeration procedure herein proposed.
Compute all permutations for the sequence of kinematic pair labels
For each member of the list compute the characteristic polynomial
Assign to each graph edge a label and form the list of all potential kinematic chains
Remove from the list the isomorphic kinematic chains
Figure 5: Flow chart of the enumeration program
9
Table 1: Structural shapes of the kinematic elements
Kinematic pair Screw Revolute Prismatic
Tubular part
Symbol
Hollow part
Symbol
Internal screw Shaft Piston
S+ P+ P+
External screw Hole Cylinder
S− P− P−
Table 2: Combination of kinematic elements for the links of a kinematic chain with screw, revolute and prismatic pairs
7
a
b
c
a1 :R+ S + a2 :R+ S − a3 :R− S + a4 :R− S −
b5 :S + P + b6 :S + P − b7 :S − P + b8 :S − P −
c9 :P + R+ c10 :P + R− c11 :P − R+ c12 :P − R−
Enumeration of planar kinematic chains with 1 d.o.f., 3 links and 1 screw pair
As shown in Figure 4a, for λ = 2, the simplest kinematic chain is composed of 3 links and 3 kinematic pairs (R=Revolute, P =Prismatic, S=Screw) with 1 d.o.f. During enumeration one can make a distinction between a hollow or tubular shape of the kinematic elements forming the kinematic pairs [2, 3]. In particular, Table 1 summarizes all the possible cases of elements shape. For each of the component links a, b and c, there are four combinations of kinematic elements. Hence, there are twelve morphologically different links. These are listed in Table 2 and shown in Figure 6. If the configurations shown in Figure 6 are not considered to be equivalent their combination results into eight non isomorphic kinematic chains, as specified in Table 3. If we consider not equivalent the structural shapes listed in Figure 6, their combination results into eight non isomorphic kinematic chains, as specified in Table 3.
8
Enumeration of kinematic chains with 1 d.o.f., four links and screw pairs
We assume that in the kinematic chain there are two screw pairs, two prismatic pairs and only one revolute pair. The computer program has been developed according to the flow-chart shown in Figure 5. There are
10
b5
a1 R+
S+
S+
P+
S-
S+
a3
R-
R+
c10
P-
R-
P+
b7
S-
S+
P+
c11
P-
R+
b8
a4 R-
P+
b6
a2 R+
c9
S-
c12 P-
S-
P-
R-
Figure 6: Structural shape of kinematic elements in kinematic chains with screw, revolute and prismatic pairs (Adapted from [2])
Table 3: List of non isomorphic kinematic chains with three links
Kinematic chain I II III IV
Links a1 a1 a2 a2
b7 b8 b5 b6
Kinematic chain c12 c10 c12 c10
V VI VII VIII
11
Links a3 a3 a4 a4
b7 b8 b5 b6
c11 c9 c11 c9
thirty possible permutations of the list {A} ≡ {SSP P R}. The permutations correspond to the following non-isomorphic kinematic chains: {A}1 ≡ {SP P SR} ,
{A}2 ≡ {SP SP R} ,
{A}3 ≡ {SP SRP } ,
{A}4 ≡ {P SP SR} ,
{A}5 ≡ {P SP RS} ,
{A}6 ≡ {P SSP R} ,
{A}7 ≡ {RSP P S} ,
{A}8 ≡ {RSP SP } ,
{A}9 ≡ {RSSP P } .
The isomorphis test based on the characteristic polynomial has been applied to the labeled adjacency matrix. In the case of permutation {A}1 in the list, this matrix takes the form:
0 S S 0 [M ] = P P R S
P P 0 0
R S . 0 0
(35)
Kinematic chains with binary links adjacent to two kinematic pairs ( see kinematic chains No.1 and 9 of Figure 7) are discarded. In this case, an uncontrolled degree-of-freedom along the axis of the kinematic pairs will occur.
b
b b a
c
a
c
a d
d
d
c
1
2
3
b c c
c
d
b
d
a
a
a
d b
4
5 d
c a
c
6
b
b
a
d
a
c
d
b
7
8
9
Figure 7: Kinematic chains PPSSR of 1 d.o.f. 4 links mechanisms With reference to Figure 7, the remaining kinematic chains can be split into two categories: 12
e
e
c
b
c
b
c b
a
d
a d
1
a
e
d
2
3
e a
e
a b
b c d
c
d
4
5
Figure 8: Five links kinematic chains PPSSSR 2 d.o.f. • those with relative displacement between two links proportional to the pitch difference of screw pairs i.e. (2,3,6,7); • those not satisfying the previous criterion i.e. (4,5,8), and thus of minor practical interest , will be not discussed further. The mechanisms atlases [1, 10, 11, 12, 13, 14, 3] mainly catalog kinematic chains with 3 links and often include kinematic chain No.7. The remaining three kinematic chains i.e. (2,3,6) appear to be novel.
9
Enumeration of kinematic chains with 2 d.o.f., 5 links and screw pairs
We assume that in the kinematic chain there are one revolute pair, two prismatic pairs and three screw pairs. The computer program developed according to the flow-chart shown in Figure 5 enumerates sixty possible permutations of the list {A} ≡ {SSSP P R}. However, only the following six permutations can be associated to non isomorphic labeling of the chain: {A}1 ≡ {SP P SRS} ,
{A}2 ≡ {SP P SSR} ,
{A}3 ≡ {SP P RSS} ,
{A}4 ≡ {SP SRP S} ,
{A}5 ≡ {SP SRSP } ,
{A}6 ≡ {SSP RP S}
All the kinematic chains have been depicted in Figure 8, with the exception of {A}6 . In this case a binary link is adjacent to two prismatic pairs.
10 Enumeration of mechanisms Once all these kinematic chains have been enumerated, it is interesting to recognize the non-isomorphic mechanisms which can be obtained. 13
Generally, the number of non-isomorphic mechanisms is different from the number of links. Most of contributions concentrate on the enumeration of kinematic chains instead of mechanisms. Moreover, some mechanisms isomorphism tests are limited to non labelled graphs. In the present investigation, the isomorphism test of Mruthyunjaya e Raghavan [9] has been modified and applied to the labelled mechanism graph adjacency matrix. With reference to Figure 7, all kinematic inversions of kinematic chains No. 2, No. 3 and No. 6 are not isomorphic. This is true also for kinematic chain No.7, with one exception i.e. the mechanisms obtained by assigning links b and d to the frame are isomorphic.
11 Kinematic analysis of differential screw mechanisms The method proposed herein can be considered as an extension of the loop closure method. Let us denote by - i and j the links connected by the kinematic pair; - dij the major screw diameter; - tij the relative linear displacement of link i with respect link j; - αij helix angle; - φij relative angular displacement of link i with respect to j. The following default values must be assigned to tij and φij : • tij = 0 if i and j are connected through a revolute pair; • φij = 0 if i and j are connected through a prismatic pair; d
• tij = φij 2ij tan αij if i and j are connected through a screw pair. The convention to choose the algebraic sign of αij is outlined in Figure 9. If we denote with K the set of kinematic pairs forming the generic independent loop, and with k = {ij} the generic element of the set, then the following expressions must hold: X tk = 0 , (36a) ∀k∈K
X
φk = 0 .
(36b)
∀k∈K
By writing equations (36) for a set of independent loops one obtains a system of linear equations. Once the input motions are prescribed, the solution of the system gives the finite motions of all the members. The simplest differential screw and its graph are shown in Figure 10. In this case there is only one loop and the set of edges is as follows K = (1, 3)S (3, 2)S (2, 1)R 14
αij > 0 j
j tij
i
tij
i
φij
φij
αij < 0
Figure 9: Algebraic sign convention
2
3
3
S
1
S
R
2
1
Figure 10: Differential screw mechanisms and its graph Since t21 = 0, the system (36) is characterized by:
φ12 +
2t23 d23 tan α23
t23 + t31 = 0 2t31 + =0 d31 tan α31
(37a) (37b)
Solving the system of equations (37) one obtains φ12 d23 d31 tan α23 tan α31 , 2 d31 tan α31 − d23 tan α23 = −t23 .
t23 =
(38a)
t31
(38b)
A more complex example is given by the analysis of the differential screw mechanism depicted in Figure 11. A set of independent loops is K1 ≡ (2, 1)R (1, 5)P (5, 4)S (4, 2)S K2 ≡ (1, 5)P (5, 3)S (3, 2)S (2, 1)R K3 ≡ (1, 5)P (5, 4)S (4, 3)P (3, 2)S (2, 1)R 15
4
2
5
1
P
R 2
3
S
5
S
S
S 3
P
4
1
Figure 11: Differential screw mechanisms and its graph (After [5]) In this case, (36) can be rewritten in the form t24 + t45 + t51 = 0
(39a)
t15 + t53 + t32 = 0
(39b)
t15 − t45 + t43 + t32 t54 t24 − φ12 + h24 h54 t32 t53 φ12 − − h32 h53 t54 t32 φ12 − − h54 h32
=0
(39c)
=0
(39d)
=0
(39e)
=0
(39f)
where for any i and j hij =
dij tan αij . 2
Solving (39) one obtains h32 (h53 − h54 ) h24 + h53 − h54 − h32 (h32 − h24 ) (h54 − h53 ) = φ12 h24 + h53 − h54 − h32 h53 (h24 − h32 ) = φ12 h24 + h53 − h54 − h32 h54 (h24 − h32) = φ12 h24 + h53 − h54 − h32 h24 (h54 − h53 ) = φ12 h24 + h53 − h54 − h32 h23 h53 − h54 h32 = φ12 h24 + h53 − h54 − h32
t32 = φ12 t43 t53 t54 t24 t15
this last equation is the same as the one obtained by R. Kraus [5]. 16
(40)
βij
αij
Figure 12: Nomenclature
12 Static analysis Since the mobility number is only two, the static analysis can be simplified. Considering ideal geometry, the analysis is carried out under the following assumptions: • for each link only two equilibrium equations are applied (rotation and translation along the screw axis); • the prismatic pair exerts only a reaction torque; • the revolute pair exerts only one force component along the axis of the pair. To the equilibrium equations, for each screw pair, one must append the relationship between the imposed torque Mij exerted on the screw and force Xij directed along the axis of the screw Mij − fij Xij = 0
(41)
with fij defined as follows fij =
pij sin αij cos βij + µ cos2 αij 2π sin αij cos βij − µ sin2 αij
(42)
where pij is the pitch of the screw, µ is the friction coefficient and, as shown in Figure 12, βij is the angle between the normal to the helicoid surface and the screw axis. When the force Xij is imposed, then the previous expression is modified as follows fij =
pij sin αij cos βij − µ cos2 αij . 2π sin αij cos βij + µ sin2 αij
17
(43)
Finally, the following equalities Xij = −Xji
(44)
Mij = −Mji
(45)
are also appended to the system of equations. M13 M23 M32
P X13
Mx
X23 X32
X12 Figure 13: Free-body diagrams of the differential screw With reference to Figure 13, which includes the free-body diagrams of the screw mechanism depicted in Figure 10, the following equations can be written
Mx + M23 = 0 X12 + X23 = 0 X32 + X13 + P = 0 M32 + M13 = 0 M23 − f23 X23 = 0 M13 − f13 X13 = 0 M32 + M23 = 0 X23 + X32 = 0 where Mx is the driving torque and P the resisting force. Solving the previous system of linear equations one obtains f13 f23 P f13 + f23 f23 P =− f13 + f23 f13 =− P f13 + f23
Mx = M13 = −M23 = − X13 X32 = X12 = −X23
18
Mx
M32
M42
M23 M53 M
M24
43
X12 X32
X42
X23
X53
M34 M54
M35
M45
X24 X54
X35
X45
M15 P
Figure 14: Free-body diagrams of a differential screw Similarly, with reference to Figure 14, which includes the free-body diagrams of the screw mechanism depicted in Figure 11, the following equations are deduced Mx + M32 + M42 = 0 X12 + X32 + X42 = 0 M23 + M53 + M43 = 0 X23 + X53 = 0 M24 + M34 + M54 = 0 X24 + X54 = 0 M35 + M45 + M15 = 0 X35 + X45 + P = 0 M23 + f23 X23 = 0 M24 + f24 X24 = 0 M35 + f35 X35 = 0 M45 + f45 X45 = 0 M23 + M32 = 0 M24 + M42 = 0 M35 + M53 = 0 M45 + M54 = 0 M43 + M34 = 0 X23 + X32 = 0 X24 + X42 = 0 X35 + X53 = 0 X45 + X54 = 0
19
Solving the previous system of linear equations one obtains f45 f23 − f35 f24 Mx = P f45 − f24 + f23 − f35 f24 − f45 X23 = −X32 = X35 = −X53 = P f45 − f24 + f23 − f35 f35 − f23 P X24 = −X42 = X45 = −X54 = f45 − f24 + f23 − f35 (f45 − f24 ) f23 P M23 = −M32 = f45 − f24 + f23 − f35 (f23 − f35 ) f24 M24 = −M42 = P f45 − f24 + f23 − f35 (f45 − f24 ) f35 P M35 = −M53 = f45 − f24 + f23 − f35 (f23 − f35 ) f45 M45 = −M54 = P f45 − f24 + f23 − f35 (f23 − f35 ) (f24 − f45 ) P M43 = −M34 = f45 − f24 + f23 − f35
13 Mechanical efficiency The mechanical efficiency of a differential screw mechanisms can be computed as the ratio Lu η= (46) Lm between the work Lm made by imposed generalized forces and work Lu made by the resisting generalized forces. For the screw mechanism depicted in Figure 11, assuming Mx is the torque imposed and P is the resisting force, the expression of the mechanical efficiency is as follows: η=
P · t51 (h23 h53 − h54 h32 ) (f45 − f24 + f23 − f35 ) = . Mx · φ12 (f23 f45 − f35 f24 ) (h24 + h53 − h54 − h32 )
(47)
Similarly, for the screw mechanism depicted in Figure 13, assuming Mx is the torque imposed and P is the resisting force (forward motion), the expression of the mechanical efficiency is as follows: η=
d23 d31 tan α23 tan α31 (f13 + f23 ) P · t31 = , Mx · φ12 2f13 f23 (d23 tan α23 − d31 tan α31 )
(48)
where fij is computed according to formula (42). When P is the force imposed (backward motion), then the values of fij must be computed according to formula (43) and the previous expressions of the efficiency are, respectively, as follows Mx · φ12 ηb = , (49) P · t51 Mx · φ12 . (50) ηb = P · t31 When ηb ≤ 0, backward motion under the influence of load P only cannot occur and the mechanism is self-locking. 20
14 Conclusions In this paper, after a review of some basic kinematic structure relationships derived by R. Kraus [4, 5], kinematic chains with 4 and 5 links and 1 degree-of-freedom (dof) and screw pairs have been enumerated. The absence of isomorphism has been checked using the characteristic polynomial method whose reliability has been tested for kinematic chains fewer than eight links. Once obtained the set of kinematic chains, all the nonisomorphic mechanisms were enumerated using the criterion of T.S. Mruthyunjaya and M.R. Raghavan. For completeness, the paper includes also novel graph-based methods for kinematic, static and mechanical efficiency analysis of mechanisms with screw pairs and reduced mobility parameter.
References [1] Reuleaux, F., Kinematics of Machinery, Translated by A.B.W. Kennedy, Reprint of Dover Publications, New York, 1964. [2] Jahr, W., Knechtel, P., Grundz¨uge der Getriebelehre, Dr. Max J¨anecke, Verlagbuchshandlung, Leipzig, 1930, pp.140-166. [3] Jensen, P.W., Classical and Modern Mechanisms for Engineers and Inventors, Marcel Dekker, New York, 1991, pp.328-347. [4] Kraus, R., Getriebelehre, Verlag Technik, Berlin 1951, pp.20-41. [5] Kraus, R., Gr¨undlagen des Systematischen Getriebeaufbaus, Verlag Technik, Berlin 1952, pp.2732, 54-66. [6] Beyer, R., Kinematisch-getriebeanalytisches Praktikum, Springer Verlag, Berlin, 1958, pp.125136. [7] Beyer, R., Bewegungsverh¨altnisse und Kraftwirkungen im dreigliedrigen gleichachsigen Schraubengetriebe mit drei Schraubenpaaren, Konstruktion, 3 Jg., 1951, pp.174-178. [8] Beyer, R., Zur Geometrie und Statik des Differentialschraubengetriebes, Feinwerktechnik, 1950, pp. 200-202 [9] Mruthyunjaya, T.S., Raghavan, M.R., Computer-Aided Analysis of the Structure of Kinematic Chains, Mechanism and Machine Theory, vol.19, 1984, pp.357-368. [10] Brown, H.T., Five Hundred and Seven Mechanical Movements, 1868 (Mechanism No.266). [11] Malavasi, C., 750 Meccanismi, Ed. Ulrico Hoepli, Milano, 1946, p.113 (in italian) (see Mechanism No.301). [12] Barber, T.W., The Engineers Sketch Book, Chemical Publishing Co., New York, 1963. [13] Chironis, N.P., Sclater, N., Mechanisms and Mechanical Devices Sourcebook, McGraw-Hill, 1990, pp.191-194. [14] Artobolevsky, I.,I., Mechanisms in Modern Engineering Design, MIR Publishers, Moscow, 1975, vol.II, pp.377-442.
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