Research Article The Steiner Formula and the Polar Moment of Inertia for the Closed Planar Homothetic Motions in Complex Plane

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1 Advances in Mathematical Physics Volume 2015, Article ID , 5 pages Research Article The Steiner Formula and the Polar Moment of Inertia for the Closed Planar Homothetic Motions in Complex Plane Ayhan Tutar and Onder Sener DepartmentofMathematics,OndokuzMayisUniversity,Kurupelit,55139Samsun,Turkey Correspondence should be addressed to Ayhan Tutar; atutar@omu.edu.tr Received 29 December 2014; Accepted 23 February 2015 Academic Editor: John D. Clayton Copyright 2015 A. Tutar and O. Sener. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Steiner area formula and the polar moment of inertia were expressed during one-parameter closed planar homothetic motions in complex plane. The Steiner point or Steiner normal concepts were described according to whether rotation number was different from zero or equal to zero, respectively. The moving pole point was given with its components and its relation between Steiner point or Steiner normal was specified. The sagittal motion of a winch was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of winch and the moving arm of winch. The results obtained in the second section of this study were applied for this motion. 1. Introduction For a geometrical object rolling on a line and making a complete turn, some properties of the area of a path of a point were given by [1]. The Steiner area formula and the Holditch theorem during one-parameter closed planar homothetic motions were expressed by [2]. We calculated the expression of the Steiner formula relative to the moving coordinate system under one-parameter closed planar homothetic motions in complex plane. If the points of the moving plane which enclosethesamearealieonacircle,thenthecentreofthis circle is called the Steiner point (h = 1)[3, 4]. If these pointslieonaline,weusesteinernormalinsteadofsteiner point. Then we obtained the moving pole point for the closed planar homothetic motions. We dealt with the polar moment of inertia of a path generated by a closed planar homothetic motion. Furthermore, we expressed the relation between the area enclosed by a path and the polar moment of inertia. As an example, the sagittal motion of a winch which is described by a double hinge being fixed and moving was considered. The Steiner area formula, the moving pole point, and the polarmomentofinertiawerecalculatedforthismotion. Moreover, the relation between the Steiner formula and the polar moment of inertia was expressed. 2. Closed Homothetic Motions in Complex Plane We consider one-parameter closed planar homothetic motion between two reference systems: the fixed E and the moving E, with their origins (O, O ) and orientations in complexplane.then,wetakeintoaccountmotionrelativeto the fixed coordinate system (direct motion). By taking displacement vectors OO =Uand O O=U and the total angle of rotation α(t), the motion defined by the transformation X (t) =h(t) Xe iα(t) +U (t) (1) is called one-parameter closed planar homothetic motion and denoted by E/E,wherehisahomothetic scale of the motion E/E and X and X are the position vectors with respect to the moving and fixed rectangular coordinate systems of a point X E, respectively. The homothetic scale h and the vectors X and U, U are continuously differentiable functions of a real parameter t. In (1), X (t) is the trajectory with respect to the fixed system of a point X belonging to the moving system. If we replace U = Ue iα(t) in (1),themotioncanbewrittenas X (t) = (h (t) X U(t)) e iα(t). (2)

2 2 Advances in Mathematical Physics The coordinates of the above equation are The following expressions are used in (11): X (t) =x 1 (t) +ix 2 (t), U (t) =u 1 (t) +iu 2 (t), X=x 1 +ix 2, U(t) =u 1 (t) +iu 2 (t). Using these coordinates, we can write (3) ( 2hu 1 dα hdu 2 +u 2 dh) = a, ( 2hu 2 dα + hdu 1 u 1 dh) = b, (12) x 1 (t) +ix 2 (t) =[(h(t) x 1 u 1 )+i(h(t) x 2 u 2 )] (cos α (t) +isin α (t)). From (4), the components of X (t) may be given as x 1 (t) = cos (α (t)) (h (t) x 1 u 1 ) sin (α (t)) (h (t) x 2 u 2 ), x 2 (t) = sin (α (t)) (h (t) x 1 u 1 )+cos (α (t)) (h (t) x 2 u 2 ). (5) Using the coordinates of (2) 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 cos (α (t)) sin (α (t)) R (t) =( ), (7) sin (α (t)) cos (α (t)) we can obtain If we differentiate (5),we have (4) (6) X (t) =R(t)(h (t) X U(t)). (8) dx 1 = sin α (hx 1 u 1 )dα+cos α(dhx 1 du 1 ) cos α (hx 2 u 2 )dα sin α(dhx 2 du 2 ), dx 2 = cos α (hx 1 u 1 )dα+sin α(dhx 1 du 1 ) sin α (hx 2 u 2 )dα+cos α(dhx 2 du 2 ). (9) {(u 2 1 +u2 2 )dα+u 1du 2 u 2 du 1 }=c. The scalar term c which is related to the trajectory of the origin of the moving system may be given as follows by taking F o := F (x 1 =0,x 2 =0): 2F o =c. (13) The coefficient m m= h 2 dα = h 2 (t 0 ) dα = h 2 (t 0 )2π] (14) with the rotationnumber ] determines whether the lines with F = const. describe circles or straight lines. If ] =0,thenwe have circles. If ] =0,thecirclesreducetostraightlines.If(12), (13),and(14) are substituted in (11),then 2(F F o )=(x 2 1 +x2 2 )m+a x 1 +b x 2 (15) can be obtained A Different Parametrization for the Integral Coefficients. Equation (8) by differentiation with respect to t yields dx =dr(hx U) +R(dhX du). (16) If X=P=( p 1 ) (the pole point) is taken, 0=dX =dr(hp U) +R(dhP du) (17) can be written. Then if U=( u 1 u 2 ) is solved from (17), dh u 1 =hp 1 + dα du 2 dα, dh u 2 =h p 1 dα + du (18) 1 dα are found. If (18) is placed in (12), 2.1. The Steiner Formula for the Homothetic Motions. The formula for the area F of a closed planar curve of the point X is given by a = ( 2h 2 p 1 dα) + ( 2hdh +hdu 2 +u 2 dh), b = ( 2h 2 dα) + (2hdhp 1 hdu 1 u 1 dh) (19) F= 1 2 (x 1 dx 2 x 2 dx 1 ). (10) If (5) and (9) are placed in (10),wehave 2F = (x 2 1 +x2 2 ) h2 dα + x 1 ( 2hu 1 dα hdu 2 +u 2 dh) can be rewritten.also (19) can be expressed separately as a:= ( 2h 2 p 1 dα), b:= ( 2h 2 dα), (20) μ 1 := ( 2hdh +hdu 2 +u 2 dh), +x 2 ( 2hu 2 dα + hdu 1 u 1 dh) μ 2 := (2hdhp 1 hdu 1 u 1 dh), (21) + {(u 2 1 +u2 2 )dα+u 1du 2 u 2 du 1 }. (11) μ=( μ 1 μ 2 ).

3 Advances in Mathematical Physics 3 Using (20) and (21),theareaformula 2(F F o )=(x 2 1 +x2 2 )m+ax 1 +bx 2 +μ 1 x 1 +μ 2 x 2 (22) is found Steiner Point or Steiner Normal for the Homothetic Motions. By taking m =0,theSteinerpointS=(s 1,s 2 ) for the closed planar homothetic motion can be written Then s j = h2 p j dα, j = 1,2. (23) h 2 dα h 2 p 1 dα = s 1 m, h 2 dα = s 2 m (24) is found. If (24) is placed in (20) and by considering (22), 2(F F o )=m(x 2 1 +x2 2 2s 1x 1 2s 2 x 2 )+μ 1 x 1 +μ 2 x 2 (25) is obtained. Equation (25) is called the Steiner area formula for the closed planar homothetic motion. By dividing this by m and by completing the squares, one obtains the equation of a circle (x 1 (s 1 μ 1 2m ))2 +(x 2 (s 2 μ 2 2m ))2 (s 1 μ 1 2m )2 (s 2 μ 2 2m )2 = 2(F F 0). m (26) All the fixed points of the moving plane which pass around equal orbit areas under the motion E/E lie on the same circle with the center M=(s 1 μ 1 2m,s 2 μ 2 2m ) (27) in the moving plane. Inthecaseofh(t) = 1, sinceμ 1 =μ 2 =0,thepointM and the Steiner point S coincide [3]. Also by taking m = 0,if it is replaced in (22),thenwehave (a+μ 1 ) x 1 + (b+μ 2 ) x 2 2(F F 0 ) =0. (28) Equation (28) is a straight line. If no complete loop occurs, then η = 0 and the circles are reduced to straight lines, in other words, to a circle whose center lies at infinity. The normal to the lines of equal areas in (28) is given by which is called the Steiner normal [5]. n=( a+μ 1 b+μ 2 ) (29) 2.3. The Moving Pole Point for the Homothetic Motions. Using (18),ifP=( p 1 ) is solved, then the pole point P of the motion p 1 = dh (du 1 u 2 dα) + hdα (du 2 +u 1 dα) (dh) 2 +h 2 (dα) 2, = dh (du 2 +u 1 dα) hdα (du 1 u 2 dα) (dh) 2 +h 2 (dα) 2 (30) is obtained. For m =0,using(14) and (23),wearriveattherelationin (24) between the Steiner point and the pole point. For m=0,using(20) and (29), we arrive at the relation between the Steiner normal and the pole point as follows: ( a 2 h 2 p 1 dα b )=( )=n μ. (31) 2 h 2 dα 2.4. The Polar Moments of Inertia for the Homothetic Motions. The polar moments of inertia T symbolizeapathforclosed homothetic motions. We find a formula by using T, m,andn in this section and we arrive at the relation between the polar moments of inertia T and the formula of area F (see(37)). A relation between the Steiner formula and the polar moment of inertia around the pole for a moment was given by [6]. Müller [3]alsodemonstratedarelationtothepolar moment of inertia around the origin, while Tölke [7] inspected the same relation for closed functions and Kuruoğlu et al. [8] generalized Müller s results for homothetic motion. If we use α as a parameter, we need to calculate T= (x 1 along the path of X.Then,using(5), 2 +x 2 2 )dα (32) T=(x 2 1 +x2 2 )m+x 1 ( 2hu 1 dα) (33) +x 2 ( 2hu 2 dα) + (u 2 1 +u2 2 )dα is obtained. We need to calculate the polar moments of inertia of the origin of the moving system; therefore T o =T(x 1 =0,x 2 = 0);oneobtains If (34) is placed in (33), T o = (u 2 1 +u2 2 )dα. (34) T T o =(x 2 1 +x2 2 )m+x 1 ( 2hu 1 dα) + x 2 ( 2hu 2 dα) (35) canbewritten.alsoif(18) is placed in (35), T T o =(x 2 1 +x2 2 )m+x 1 ( 2h 2 p 1 dα 2hdh +2hdu 2 ) +x 2 ( 2h 2 dα + 2hdhp 1 2hdu 1 ) (36)

4 4 Advances in Mathematical Physics x 2 l L x 1 Figure 1: The arms of winch as a double hinge. is obtained and by considering (22) and (36) together, we arrive at the relation between the polar moments of inertia andtheformulafortheareabelow: T T o =2(F F o )+x 1 (hdu 2 u 2 dh) x 2 k x 1 +x 2 ( hdu 1 +u 1 dh). 3. Application: The Motion of the Winch (37) In the previous sections we emphasized three concepts: geometrical objects as the Steiner point or the Steiner normal, the pole point, and the polar moments of inertia for closed homothetic motions in complex plane. In this section, we want to visualize the experimentally measured motion with these objects. Accordingly, we consider these characteristic directions for this motion. We will show how the kinematical objects which are used in the previous sections can be applied. In the study by Dathe and Gezzi [5],theyconsideredhumangaitinplanar motions. As an example, we have chosen the sagittal part of the movement of the winch at motion. We have chosen the winch, because the arm of winch can extend or retract during one-parameter closed planar homothetic motion. The motion of winch has a double hinge and a double hinge means that it has two systems, a fixed arm and a moving arm of winch (Figure 1). There is a control panel of winch at the origin of fixed system. L arm can extend or retract by h parameter. By taking cos (l (t) k(t)) sin (l (t) k(t)) R (t) =( sin (l (t) k(t)) cos (l (t) k(t)) ), (39) U L cos (l (t)) (t) =( L sin (l (t)) ), we have X (t) =h(t) R (t) X+U (t). (40) Also we know that U = RU. Therefore, U (t) =( u 1 (t) cos (k (t)) )=( L u 2 (t) L sin (k (t)) ) (41) canbewritten.sothedoublehingemaybewrittenas x 1 (t) = cos (l (t) k(t)) (h (t) x 1 +Lcos (k)) sin (l (t) k(t)) (h (t) x 2 +Lsin (k)), x 2 (t) = sin (l (t) k(t)) (h (t) x 1 +Lcos (k)) + cos (l (t) k(t)) (h (t) x 2 +Lsin (k)). (42) We begin by calculating the time derivative of (42). Inthis way, we obtain the velocities x1 (t), x2 (t), whichhavetobe inserted into (10): x 1 x 2 x 2 x 1 =(h 2 (x 2 1 +x2 2 )+L2 )( l (t) k (t)) +x 1 (2hL cos (k (t)) ( l (t) k (t)) +hlcos (k (t)) k (t) Ldhsin (k (t))) +x 2 (2hL sin (k (t)) ( l (t) k (t)) +L 2 k (t). +hlsin (k (t)) k (t) +Ldhcos (k (t))) (43) We now integrate the previous equation using periodic boundary conditions by assuming the integrands as periodic functions. The periodicity of f implies that integrals of the fdt = f F 1 =0.Asaresult following types vanish, df = F 1 of this, some of the integrals of (43) arenotequaltozeroand we finally obtain a simplified expression for the area 3.1. The Mathematical Model. We start by writing the equations of the double hinge in Cartesian coordinates. Then we define, using the condition m=0, the Steiner normal and the total angle in relation to the double hinge. By taking displacement vectors OO =Uand O O=U and the total angle of rotation l k = α,themotioncanbe defined by the transformation X (t) =h(t) Xe i(l(t) k(t)) +U (t). (38) 2F = x 1 ( 2Lh cos k( l k) dt + L(hcos k k dhsin k) dt) +x 2 ( 2Lh sin k( l k) dt + L(hsin k k+dhcos k) dt). (44)

5 Advances in Mathematical Physics 5 We may have the following expressions from (44): ( 2Lh cos k( l =a, ( 2Lh sin k( l k) dt + k) dt + L(hcos k k dhsin k) dt) L(hsin k k+dhcos k) dt) =b. (45) Differentiating (41) with respect to t andthenusingtheresult in (45),weobtain(12) for application. In Section 2.1.1,using(18), a = b = ( 2h 2 p 1 dα) a ( 2h 2 dα) b + ( 2hdh +hdu 2 +u 2 dh), μ 1 + ( 2hdhp 1 +hdu 1 +u 1 dh) μ 2 (46) arefoundandwehaveastraightlinebelow: 2F = (a+μ 1 ) x 1 + (b+μ 2 ) x 2. (47) In this case, we have the Steiner normal n=( a+μ 1 b+μ 2 ) ( {2h cos k( l k) + (h cos k k dhsin k)} dt) t =L( 1 ). ( {2h sin k( l k) + (h sin k k+dhcos k)} dt) (48) 3.2. The Moving Pole Point of the Winch Motion. If (41) is replaced in (30), the pole point P=( p 1 ) with the components p 1 = dh (L sin k l) h ( l k) (L cos k l) (dh) 2 +h 2 ( l k) 2, = dh ( L cos k l) h ( l k) (L sin k l) (dh) 2 +h 2 ( l k) 2 is obtained and P=( p 1 ) = L l (dh) 2 +h 2 ( l (49) k) 2 ( dh sin k h( l k) cos k dh cos k h( l k) sin k ) (50) 3.3. The Polar Moments of Inertia of the Winch Motion. Using (32) and (42),if(41) is replaced in (33), T=x 1 2hL cos k( l k) dt + x 2 2hL sin k( l k) dt (51) is obtained. By considering (46), (47), and(51) together, we arrive at the relation between the polar moments of inertia and the formula for the area below: T=2F+x 1 L ( h cos k k+dhsin k) Conflict of Interests x 2 L (h sin k k+dhcos k). (52) The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment This study is supported by Ondokuz Mayıs University (Project no. PYO.FEN ). References [1] J. Steiner, Von dem Krümmungs-Schwerpuncte ebener Curven, Journal für die Reine und Angewandte Mathematik, vol. 1840,no.21,pp.33 63,1840. [2] A. Tutar and N. Kuruoğlu, The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics, Mechanism and Machine Theory,vol.34,no.1,pp. 1 6, [3] H. R. Müller, Verallgemeinerung einer formel von steiner, Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft,vol.29,pp ,1978. [4] H. R. Müller, Über Trägheitsmomente bei Steinerscher Massenbelegung, Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, vol. 29, pp , [5] H. Dathe and R. Gezzi, Characteristic directions of closed planar motions, Zeitschrift für Angewandte Mathematik und Mechanik,vol.92,no.9,pp ,2012. [6] W. Blaschke and H. R. Müller, Ebene Kinematik,R.Oldenbourg, Munich, Germany, [7] J. Tölke, Steiner-Formein für die Bahnflachen geschlossener Aquiaffinbewegungen, Sitzungsber, Österreichische Akademie der Wissenschaften,vol.187,no.8 10,pp ,1978. [8] N. Kuruoğlu, M. Düldül, and A. Tutar, Generalization of Steiner formula for the homothetic motions on the planar kinematics, Applied Mathematics and Mechanics,vol.24,no.8, pp , canbewritten.alsousing(46) and (48),wereachtherelation between the Steiner normal and the pole point (31).

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