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

Abstract

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.

The Kinetic Energy Formula for the Closed Planar Homothetic Motions in Complex Plane

“…Also we define using the condition m = 0 (Eq. (15) of [8]), the Steiner line and the total angle in relation to the double hinge. By taking displacement vectors OO ′ = U ′ and O ′ O = U , the total angle of rotation l − k = a, the motion defined by the transformation…”

“…Then If Eq. (39) of [8] and Eq. (24) are replaced in calculating datȧ x 2 1 +ẋ 2 2 and in Section 2, by using the parameter dt → da,ȧ → a ′ = da da = 1, → : and Eqs.…”

“…(24) are replaced in calculating datȧ x 2 1 +ẋ 2 2 and in Section 2, by using the parameter dt → da,ȧ → a ′ = da da = 1, → : and Eqs. (46) and (47) of [8] are replaced in Eq. (27),…”

“…Dathe and Gezzi expressed the Steiner area formula and the polar moment of inertia for the closed planar motions [1]. The study reported by Dathe and Gezzi [1] was investigated during one-parameter closed planar homothetic motions in complex plane by Tutar and Sener [8]. In the case of the homothetic scale h ≡ 1 the results given by [1] were obtained as a special case.…”

“…Eq. (18) of [8] is as below: [8] is replaced at coefficients of x 1 and x 2 in Eq. (15), respectively and by calculating necessary operations,…”