x 1 ( t ) + i x 2 ( t ) = 1 h ( t ) [ ( x 1 u 1 ( t ) ) + i ( x 2 u 2 ( t ) ) ] ( cos α ( t ) i s i n α ( t ) ) . (2)

From Equation (2), the components of X ( t ) may be given as

x 1 ( t ) = 1 h ( t ) [ cos ( α ( t ) ) ( x 1 u 1 ( t ) ) + sin ( α ( t ) ) ( x 2 u 2 ( t ) ) ] x 2 ( t ) = 1 h ( t ) [ sin ( α ( t ) ) ( x 1 u 1 ( t ) ) + cos ( α ( t ) ) ( x 2 u 2 ( t ) ) ] } (3)

If we show the coordinates of the Equation (1) as

X ( t ) = ( x 1 ( t ) x 2 ( t ) ) , X = ( x 1 x 2 ) , U ( t ) = ( u 1 ( t ) u 2 ( t ) ) , U ( t ) = ( u 1 ( t ) u 2 ( t ) )

and rotation matrix

R ( t ) = ( cos ( α ( t ) ) s i n ( α ( t ) ) s i n ( α ( t ) ) cos ( α ( t ) ) ) , (4)

we can obtain

X ( t ) = 1 h ( t ) ( R ( t ) ) T ( X U ( t ) ) . (5)

If Equation (3) differentiated, we have

d x 1 = d h h 2 [ cos α ( x 1 u 1 ) + sin α ( x 2 u 2 ) ] + 1 h [ ( cos α ) d u 1 ( x 1 u 1 ) ( sin α ) d α ( s i n α ) d u 2 + ( x 2 u 2 ) ( cos α ) d α ] d x 2 = d h h 2 [ sin α ( x 1 u 1 ) + cos α ( x 2 u 2 ) ] + 1 h [ ( s i n α ) d u 1 ( x 1 u 1 ) ( cos α ) d α ( c o s α ) d u 2 ( x 2 u 2 ) ( sin α ) d α ] } (6)

2.1. The Steiner Formula for the Homothetic Inverse Motions

The formula for the area F of a closed planar curve of the point X is given by

F = 1 2 ( x 1 d x 2 x 2 d x 1 ) . (7)

If Equations (3) and (6) are replaced in Equation (7),

2 F = ( x 1 2 + x 2 2 ) 1 h 2 d α + x 1 ( 2 1 h 2 u 1 d α 1 h 2 d u 2 ) + x 2 ( 2 1 h 2 u 2 d α + 1 h 2 d u 1 ) { 1 h 2 ( u 1 2 + u 2 2 ) d α 1 h 2 ( u 1 d u 2 u 2 d u 1 ) }
(8)

is obtained. The integral coefficients in Equation (8) can be shown as

( 2 1 h 2 u 1 d α 1 h 2 d u 2 ) = a ( 2 1 h 2 u 2 d α + 1 h 2 d u 1 ) = b { 1 h 2 ( u 1 2 + u 2 2 ) d α 1 h 2 ( u 1 d u 2 u 2 d u 1 ) } = c } (9)

If we show the trajectory of the orjin of the fixed system by F o = F ( x 1 = 0 , x 2 = 0 ) , we can say

2 F o = c . (10)

The coefficient m is defined by

m = 1 h 2 d α = 1 h 2 ( t 0 ) d α = 1 h 2 ( t 0 ) 2 π ν (11)

with the rotation number ν establishes whether the lines with F = c o n s t . describing circles or straight lines. If ν 0 , then we have circles. If ν = 0 , the circles reduce to straight lines. If Equations (9), (10) and (11) are substituted in Equation (8), then

2 ( F F o ) = ( x 1 2 + x 2 2 ) m + a x 1 + b x 2 (12)

can be written.

A Different Parametrization for the Integral Coefficients

Equation (5) by differentiation with respect to t yields

d X = 1 h d ( R T ) ( X U ) 1 h R T d U d h h 2 R T ( X U )

and if we use X = P = ( p 1 p 2 ) for the pole point, we can write

0 = d X = 1 h d ( R T ) ( P U ) 1 h R T d U d h h 2 R T ( P U ) (13)

Then if U = ( u 1 athvariant="script"> l ˙ cos l ) d t = a t 1 t 2 ( 2 1 h 2 L sin l ( l ˙ k ˙ ) 1 h 2 L l ˙ sin l ) d t = b } (40)

Differentiating Equation (36) with respect to t and then replacing both of them in Equation (40), Equation (9) is found for application.

In Section (2.1.1), using Equation (14),

a = t 1 t 2 ( 2 1 h 2 p 1 d α ) a + t 1 t 2 ( 2 d α h 3 ( d α ) 2 + h ( d h ) 2 ( d h d u 1 + h d α d u 2 ) 1 h 2 d u 2 ) μ 1 b = t 1 t 2 ( 2 1 h 2 p 2 d α ) b + t 1 t 2 ( 2 d α h 3 ( d α ) 2 + h ( d h ) 2 ( d h d u 2 h d α d u 1 ) 1 h 2 d u 1 ) μ 2 (41)

is found and we have a straight line below:

2 F = ( a + μ 1 ) x 1 + ( b + μ 2 ) x 2 . (42)

In this case, we have the Steiner normal

n = ( a + μ 1 b + μ 2 ) = L ( t 1 t 2 ( 2 1 h 2 cos l ( l ˙ k ˙ ) 1 h 2 l ˙ cos l ) d t t 1 t 2 ( 2 1 h 2 sin l ( l ˙ k ˙ ) 1 h 2 l ˙ sin l ) d t ) (43)

3.2. The Fixed Pole Point of the Inverse Telescopic Crane Motion

If Equation (36) is replaced in Equation (26), the pole point P = ( p 1 p 2 ) with the components

p 1 = h ( d h ) 2 + h 2 ( l ˙ k ˙ ) 2 ( d h L l ˙ sin l h ( l ˙ k ˙ ) L l ˙ cos l ) + L cos l p 2 = h ( d h ) 2 + h 2 ( l ˙ k ˙ ) 2 ( d h L l ˙ cos l + h ( l ˙ k ˙ ) L l ˙ sin l ) + L sin l } (44)

is found and also by using Equations (43) and (44), we reach at the relation between the Steiner normal and the pole point (Equation (27)).

3.3. The Polar Moments of Inertia of the Inverse Telescopic Crane Motion

Using Equations (28) and (37), if Equation (36) is replaced in Equation (29)

T = x 1 ( 2 1 h 2 L ( cos l ) ( l ˙ k ˙ ) ) d t + x 2 ( 2 1 h 2 L ( sin l ) ( l ˙ k ˙ ) ) d t (45)

is found. By considering Equations (41), (42) and (45) together, we arrive at the relation between the polar moments of inertia and the formula for the area below:

T = 2 F x 1 L 1 h 2 ( cos l ) l ˙ d t x 2 L 1 h 2 ( sin l ) l ˙ d t (46)

Acknowledgements

This study is supported by Ondokuz Mayıs University (Project No. PYO. FEN. 1904. 14.019).

Cite this paper
Sener, O. and Tutar, A. (2017) On the Kinematics for the Closed Planar Homothetic Inverse Motions in Complex Plane. Journal of Applied Mathematics and Physics, 5, 1120-1129. doi: 10.4236/jamp.2017.55099.
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