MODELOWANIE INŻYNIERSKIE ISNN 1896-771X 32, s. 345-352, Gliwice 2006
KINEMATICS OF RELATIVE BODY MOTION AND PLŰCKER’S CONOID
VLADIMÍR MACHULDA
JAROMÍR ŠVÍGLER
Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia in Pilsen
Abstract. Spatial motion of rigid body is possible to describe by using the screw movement around a certain axis. In this contribution the relative motion between two bodies is analyzed. The kinematic solution is described using kinematic properties of Plűcker’s conoid. Simultaneously the classical kinematic solution of relative motion of two bodies is mentioned. Application of the presented theory to rotors of screw compressor with displacement of rotors position is shown.
1. INTRODUCTION
The axes of rotors of screw compressor are designed parallel. After bearing displacement, caused by action of force and heat on compressor housing, the parallel position of rotors changes into spatial arrangement and relative rotary motion changes into relative screw motion, twist, around certain axis, which is determined by mutual position of the axes of both rotors. For determination of position of axes of relative twist we can use kinematic properties of Plűcker’s conoid, which is presented in this contribution.
2. KINEMATICS OF RELATIVE MOTION OF TWO BODIES 2.1. Configuration of axes
Let two rigid bodies get around skew axes o2, o3 screw motion, twist. Relative position of axes of these twists is defined by transversal a and angle Σ. The position of axis of relative twist o32 is determined by transversal a2 and angle γ2. Configuration of axes is shown in Fig.1.
Fig. 1. Configuration of axes
2.2. Determination of axis of relative twist
Relative motion of two bodies can be expressed with using of classical kinematic way as equation system (1), see [1]. Transversals a2 and a3 and angles γ2 and γ3 are conjugate with transversal a and angle Σ through first two relations of equation system (1).
. sin sin
sin cos
, sin sin
, ,
2 2 2 2 2 3 3 3 3 3
2 2 3 3
2 3
2 3
γ γ
ω γ
γ ω
γ ω γ ω
γ γ
v a
v a
a a a
+
= +
= Σ
= +
= +
(1)
Gear ratio can be determined by relation
32 .
2 3 =i ω
ω (2)
With the use of equation system (1) and relation (2) we can determine position of axis of relative twist by equations (3) and (4).
Σ +
= Σ 1 cos tan sin
32 2
i
γ (3)
( )
[
3 3 2 3]
2
2 cos sin
sin
sinγ γ γ
p p a
a + −
= Σ , (4)
where p2 and p3 are parameters of screw motions.
3. PLŰCKER’S CONOID
Plűcker’s conoid, Fig. 2, is a surface that has from the kinematic point of view properties which make this surface important in mechanics. The first application of Plűcker’s conoid to dynamics was made in 1869 by Battagliny, who showed that this surface was the locus of wrench resulting from the composition of forces of varying ratio on two given straight lines, see [2].
Fig. 2. Plűcker’s conoid
3.1. Geometric properties
From the geometric point of view Plűcker’s conoid is skew straight-line surface of 3rd degree, which is determined by elliptic section k of vertical circular cylindrical surface, line d of this surface and vanishing line of first plane of projection π, see [3]. This surface has two torsal planes 1τ and 2τ and each of one contains one torsal line t. These torsal lines are perpendicular, as you can see in Fig. 3.
In coordinate system Rk ≡
(
xk,yk,zk)
we can express position of line of Plűcker’s conoid as, tan
, cos sin
ϑ ϑ ϑ
k k
k
x z
p y
=
= (5)
where parameter p is distance between torsal planes and ϑ is angle, which line of Plűcker’s conoid contains with axis xk.
Fig. 3. Geometry of Plűcker’s conoid
3.2. Kinematic properties
From the kinematic point of view Plűcker’s conoid has three important properties.
Property 1
We consider pair of kinematic screws α and β with parameters pα and pβ, which axes coalesce with coordinate axes xk and zk. These twists are called principal screw, see [2], and their axes create main lines of Plűcker’s conoid. Axes of any screw motions Θ
(
o ,Θ pΘ)
, whichare composed of screws α and β, create surface of Plűcker’s conoid. Position of axes is determined by equation
(
pα pβ)
sinϑcosϑ.yΘ = − (6)
Property 2
We can determine Plűcker’s conoid for each pair of twists which axes lie on its surface.
Consider pair of twists Θ2
(
o2, p2)
and Θ3(
o3, p3)
. Relative position of axes of these twists is determined by transversal a and angle Σ. With the use of equation system (7) we can determine parameters of principal screws and position of axes of primary twists on surface of Plűcker’s conoid.( )
( )
( )
( )
.tan 2
1
tan , 2
1
, arctan
2 1
, arctan
2 1
sin , tan
2 1
tan , sin
2 1
2 3 3
2 3 2
2 3 3
2 3 2
2 2 3 2 2
3
2 3 2 2 3 2
+
Σ
= −
−
Σ
= −
− +Σ
=
− −Σ
=
Σ
−
− +
− Σ +
=
− Σ + Σ +
−
= +
p a y p
p a y p
a p p
a p p
p p a a
p p p
p a p p
p p a
ϑ ϑ
β α
(7)
Property 3
Axis of twist, which is composed of two twists, lies on surface of the same Plűcker’s conoid as axes of these twists.
4. APPLICATION TO ROTORS OF SCREW COMPRESSOR 4.1. Simulation of bearing displacement
First of all we establish coordinate system of rotors. In this coordinate system the axis of female rotor o2 coalesces with axis z and axis of male rotor o3 is parallel and lies in plane yz in distance a. Then the position of rotors in the coordinate system of rotors is determined as
[ ]
[ ]
[
0 0 1]
., 0 0
, 0 0 0
3 2
3 2
T T T
o o
a O
O
=
≡
=
=
r r
(8)
Bearing displacement can be simulated as the displacement of axis o3 determined by x∆ ,
∆y, z∆ and subsequent turning over angles ξ and ζ, see Fig. 4. Axis o3 changes in axis o , 3∆ which is defined by point O and direction vector 3∆ or3∆
[ ]
[
sin cos sin cos cos]
.,
3 3
T T
o
z y a x O
ζ ξ ζ
ξ
ξ ⋅ ⋅
=
∆
∆ +
∆
=
∆
∆
r (9)
Relative position of axes o2 and o is determined by transversal d and angle Σ, which are 3∆ determined by equations
( )
. cos cos cos
3 ,
2 3
3 2 0 3
ζ ξ
= Σ
+ +
= + +
= ∆
∆ D D D D
O r d r k r d d o r
rr r r r r r r
(10)
Fig. 4. Position of axes and coordinate system of Plűcker’s conoid
4.2. Coordinate system of Plűcker’s conoid
For using relations (5) and (6) we must first determine position of the relative twist’s axis in the coordinate system of Plűcker’s conoid, see Fig. 3, then the use the kinematic properties of Plűcker’s conoid in this coordinate system is simple. The coordinate system has to satisfy three conditions. The first condition is that transversal d lies on axis yk. The second condition determine origin of coordinates in half of length transversal d and the last condition is
3
2 ϑ
ϑ =− . Transformation from the system of Plűcker’s conoid is defined by matrix
.
1 0
0
0 0 sin 2
cos2
2 cos 1 cos2 sin 2 cos
sin sin
2 sin 1 cos2
cos 2 sin
sin cos
2
Σ Σ
⋅ Σ Σ −
⋅
Σ −
⋅ Σ −
⋅
=
D R
R
r d
d
k
β β
β β
β β
β β
T (11)
4.3. Determination of axis of relative twist
With the use of the first three relations of equation system (7) and relation (3) we can determine parameters of principal screws of Plűcker’s conoid and angles ϑ and γ2 2.Then angle ϑ , which axis of relative twist contains with axis x32 k of the coordinate system Rk, can be expressed as
2 .
2
32 ϑ γ
ϑ = + (12)
With the use of relations (5) and (6) the position of relative twist’s axis in coordinate system of Plűcker’s conoid can be defined as
[ ]
[
cos 0 sin]
., 0 0
32 32
32
32 32
T k
T k k
o
y O
ϑ ϑ
=
=
r (13)
Now we need to transform these results to coordinate system of rotors. This transformation is expressed by equations
0 . 0
1 , 1
32 32
32 32
=
=
k RR
k RR
o o
O O
k k
r r
T T
(14)
Equations (13) and (14) solve the problem of position of relative twist’s axis in the coordinate system of rotors.
Application of the mentioned theory was implemented on the oil injected screw compressor, which is loaded by force and temperature fields. These fields make deformations of the
compressor housing that induce of bearing displacements. Numerical solution was made for next basic parameters of the screw machine: axes distance aw = 85mm, gear ratio i32 = 1,2, where 3 is male rotor and 2 is female rotor, helix angle on the rolling cylinder of both rotors γ
= 45°, length of tooth part of both rotors l = 193,8 mm. Values of mutual relative displacement of both rotor axes, caused by bearing displacements, which was obtained by numerical simulation of the housing deformations are ∆x=−2,05⋅10−5m, ∆y=7,69⋅10−5m,
m z=−6,11⋅10−8
∆ , ξ =9,08´, ζ =−21,26´, Σ=23,12´. With the use of equations (9) we concretely determine point O and direction vector 3∆ or3∆
in the coordinate system R≡
(
x,y,z)
[ ]
[
0,00264 -0,00619 0,99998]
., 10 11 , 6 10
0769 , 85 10 05 , 2
3
8 3
5 3
T
T
o O
=
⋅
−
⋅
⋅
−
=
∆
−
−
−
∆
r (15)
With the use of equation system (13) we can determine the position of relative twist’s axis in coordinate system of Plűcker’s conoid
[ ]
[
0,99999 0 -0,00031]
., 0 10 1,51 0
32
3 - 32
T k
T k
o O
=
⋅
=
r (16)
Transformation matrix defined by (11) is in this case
. 1 0
0 0
11,63614 00336
, 0 0
99999 , 0
00655 , 0 91971 , 0 0.39259 00309
, 0
0,01535 39259
, 0 91971 , 0 0,00132
−
−
R =
Rk
T (17)
With using of equation (14) we determine the position of relative twist’s axis in coordinate system of rotors
[ ]
[
0,00144 0,00337 0,99999]
., 11,64 10
7,15 10
1,67
32
3 - 2
- 32
T T
o O
−
=
⋅
⋅
=
r (18)
Parameter of twist is determined with equation
. sin cos2ϑ β 2ϑ
α p
p
pΘ = + (19)
After substitution in equation (19) the value of parameter of relative twist is .
10 57 ,
5 5 1
32
−
− ⋅
⋅
−
= m rad
p
Operating angular velocity of male rotor is in the case of screw compressor is
1 3 =575,96rad⋅s−
ω . Angular velocity of female rotor is defined by equation (2). Angular velocity of relative motion is defined by vector equation
( )
2 . 23 3 2 3
32 r r or or
r =ω −ω =ω ∆ −ω −
ω (20)
The value of angular velocity of relative motion is . s rad 96,06 -1
32 = ⋅
ω (21)
Value of velocity of relative motion can be determined with the use of parameter of relative twist. In this case the value of velocity of relative motion is
. 10
5,35 -3 1
32
⋅ −
⋅
= m s
v (22)
5. CONCLUSION
This contribution is pointed at kinematic properties of Plűcker’s conoid and using of these properties for kinematic analysis of two rigid bodies. Application was implemented on rotors of screw compressor with bearing displacement. The axis of relative motion of these rotors, which are in spatial arrangement, was determined. The relative rotary motion changes into relative twist. Results of kinematic analysis can be used for quasistatic analysis of screw compressor after bearing displacement and for determination of excitation vector by dynamic analysis. The presented theory offers comparatively easy way for solution of space problems of rigid bodies.
Acknowledgement: This research work was supported by MSM 4977751303.
REFERENCES
1. Šejvl, M.: Aplikovaná kinematika na prostorové převody. Doctoral thesis, Plzeň 1967.
2. Ball, S.: A treatise on the theory of screws. Cambridge University Press 1998, First published 1900.
3. Kadeřávek, F., Klíma, J., Kounovský, J.: Deskriptivní geometrie II. Knihovna spisů matematických a fysikálních, Volume 17, Praha 1932.
