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.

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 enclose the same area lie on a circle, then the centre of this circle is called the Steiner point () [3, 4]. If these points lie on a line, we use Steiner normal instead of Steiner 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 polar moment of inertia were calculated for this motion. 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 and the moving , with their origins and orientations in complex plane. Then, we take into account motion relative to the fixed coordinate system (direct motion).

By taking displacement vectors and and the total angle of rotation , the motion defined by the transformation is called one-parameter closed planar homothetic motion and denoted by , where is a homothetic scale of the motion and and are the position vectors with respect to the moving and fixed rectangular coordinate systems of a point , respectively. The homothetic scale and the vectors and are continuously differentiable functions of a real parameter .

In (1), is the trajectory with respect to the fixed system of a point belonging to the moving system. If we replace in (1), the motion can be written as The coordinates of the above equation are Using these coordinates, we can write From (4), the components of may be given as Using the coordinates of (2) as and rotation matrix we can obtain If we differentiate (5), we have

2.1. The Steiner Formula for the Homothetic Motions

The formula for the area of a closed planar curve of the point is given by If (5) and (9) are placed in (10), we have The following expressions are used in (11): The scalar term which is related to the trajectory of the origin of the moving system may be given as follows by taking : The coefficient with the rotation number determines whether the lines with = const. describe circles or straight lines. If , then we have circles. If , the circles reduce to straight lines. If (12), (13), and (14) are substituted in (11), then can be obtained.

2.1.1. A Different Parametrization for the Integral Coefficients

Equation (8) by differentiation with respect to yields If (the pole point) is taken, can be written. Then if is solved from (17), are found.

If (18) is placed in (12), can be rewritten. Also (19) can be expressed separately as Using (20) and (21), the area formula is found.

2.2. Steiner Point or Steiner Normal for the Homothetic Motions

By taking , the Steiner point for the closed planar homothetic motion can be written Then is found. If (24) is placed in (20) and by considering (22), is obtained. Equation (25) is called the Steiner area formula for the closed planar homothetic motion.

By dividing this by and by completing the squares, one obtains the equation of a circle All the fixed points of the moving plane which pass around equal orbit areas under the motion lie on the same circle with the center in the moving plane.

In the case of , since , the point and the Steiner point coincide [3]. Also by taking , if it is replaced in (22), then we have Equation (28) is a straight line. If no complete loop occurs, then 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].

2.3. The Moving Pole Point for the Homothetic Motions

Using (18), if is solved, then the pole point of the motion is obtained.

For , using (14) and (23), we arrive at the relation in (24) between the Steiner point and the pole point.

For , using (20) and (29), we arrive at the relation between the Steiner normal and the pole point as follows:

2.4. The Polar Moments of Inertia for the Homothetic Motions

The polar moments of inertia “” symbolize a path for closed homothetic motions. We find a formula by using , , and in this section and we arrive at the relation between the polar moments of inertia “” and the formula of area “” (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] also demonstrated a relation to the polar 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 as a parameter, we need to calculate along the path of . Then, using (5), is obtained.

We need to calculate the polar moments of inertia of the origin of the moving system; therefore ; one obtains If (34) is placed in (33), can be written. Also if (18) is placed in (35), is obtained and by considering (22) and (36) together, we arrive at the relation between the polar moments of inertia and the formula for the area below:

3. Application: The Motion of the Winch

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], they considered human gait in planar 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. “” arm can extend or retract by parameter.

Figure 1 

The arms of winch as a double hinge.

3.1. The Mathematical Model

We start by writing the equations of the double hinge in Cartesian coordinates. Then we define, using the condition , the Steiner normal and the total angle in relation to the double hinge.

By taking displacement vectors and and the total angle of rotation , the motion can be defined by the transformation By taking we have Also we know that . Therefore, can be written. So the double hinge may be written as We begin by calculating the time derivative of (42). In this way, we obtain the velocities , which have to be inserted into (10): We now integrate the previous equation using periodic boundary conditions by assuming the integrands as periodic functions. The periodicity of implies that integrals of the following types vanish, . As a result of this, some of the integrals of (43) are not equal to zero and we finally obtain a simplified expression for the area We may have the following expressions from (44): Differentiating (41) with respect to and then using the result in (45), we obtain (12) for application.

In Section 2.1.1, using (18), are found and we have a straight line below: In this case, we have the Steiner normal

3.2. The Moving Pole Point of the Winch Motion

If (41) is replaced in (30), the pole point with the components is obtained and can be written. Also using (46) and (48), we reach the relation between the Steiner normal and the pole point (31).

3.3. The Polar Moments of Inertia of the Winch Motion

Using (32) and (42), if (41) is replaced in (33), 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:

Conflict of Interests

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.1904.14.019).