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Introduction Basic Integral Functions Elliptical Plate Semi-Ellipsoid (SE) Elliptical Solid (ES) Stadium Plate Stadium Solid (SS) Applications References and Related Literature

Introduction

Three geometric groups commonly used in defining the body segments are semi-ellipsoids, elliptical solids and stadium solids. The semi-ellipsoid group consists of semi-ellipsoid, ellipsoid of revolution and sphere. The typical members of the elliptical solid group are the elliptical and circular cylinders, the truncated or normal circular and elliptical cones and the general elliptical and circular solids. The stadium solid group consists of the stadium solids and the stadium cylinders:

Group	Member Geometric Shapes
Semi-Ellipsoid (SE)	Semi-ellipsoid   Ellipsoid   Ellipsoid of revolution   Sphere
Elliptical Solid (ES)	Elliptical solid   Truncated circular cone   Elliptical Cylinder   Elliptical disk (zone)   Circular cylinder
Stadium Solid (SS)	Stadium solid   Stadium cylinder

For each geometric group, equations for the mass (m), center-of-mass (CM) location from the proximal end (g_z), and the three principal moments of inertia (I_xxx, I_yy & I_zz) are provided. A uniform density of r is assumed here. The BSP equations for the stadium solid were adapted and modified from Yeadon (1990).

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Basic Integral Functions

Let functions a(t), b(t), c(t) & d(t) be functions of t

    [1]

where a, b, c, d > 0 and 0 < t < 1. Then,

   [2]

Now, let functions a(t), b(t), c(t) & d(t) be

    [3]

where, a_o, a₁, b_o, b₁, c_o, c₁, d_o & d₁ > 0 and 0 < t < 1. Then,

    [4]

where

   [5]

A set of trigonometric integral functions were also used in deriving the BSP functions:

   [6]

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Elliptical Plate

   Figure 1

The mass and principal moments of inertia of an elliptical plate (Figure 1) is as follows:

   [7]

Note that [6] was used in deriving [7].

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Semi-Ellipsoid (SE)

   Figure 2

Let z in Figure 2 be a function of t (0 < t < 1):

    [8]

Then,

   [9]

[2] & [7] were used in deriving [9]. The BSP equations for ellipsoid, ellipsoid of revolution, and sphere can be derived from [9]:

Geometric Shape	Properties
Ellipsoid	m = 2mSE, gz = 50 %
Ellipsoid of Revolution	a = b, m = 2mSE, gz = 50 %
Sphere	a = b = c, m = 2mSE, gz = 50 %

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Elliptical Solid

    Figure 3

Let z in Figure 3 be a function of t (where 0 < t < 1):

   [10]

Then,

    [11]

[3], [4], [5] & [7] were used in deriving [11]. The BSP equations for truncated circular cone, elliptical cylinder, and circular cylinder can be also derived from [11]:

Geometric Shape	Properties
Truncated Circular Cone	ao = bo, a1 = b1
Elliptical Cylinder   Elliptical Zone (Disk)	ao = a1, bo = b1
Circular Cylinder	ao = a1 = bo = b1

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Stadium Plate

    Figure 4

The mass and moment of inertia functions of the stadium plate (Figure 4) were obtained as follows:

   [12]

[6] was used in deriving [12].

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Stadium Solid

   Figure 5

Figure 5 shows the stadium solid. Note that the directions of the axes are slightly different from Figures 2 & 3. Let z be a function of t (where 0 < t < 1):

   [13]

From [10]:

   [14]

[3], [4], [5] & [12] were used in deriving [13]. In the stadium cylinders, a_o = a₁ and r_o = r₁. If a_o = a₁ = 0, it becomes a circular solid (truncated circular cone). If a₁ = r₁ = 0, the stadium solid becomes a stadium cone. Note here that [4], [5] & [13] are presented in a slightly different form from that in Yeadon (1990). Yeadon used a & b that were defined as

   [15]

When r_o = 0, or a_o = 0, [15] causes divided-by-zero error. The equations presented in this page prevent this problem and, thus, are more flexible.

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Applications

See BSP Estimation Methods or Kwon (1996) for the modified Hanavan model and the modified Yeadon model.

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References & Related Literature

Hanavan, E. P. (1964). A mathematical model of the human body. AMRL-TR-64-102, AD-608-463. Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio.

Hatze, H. (1980). A mathematical model for the computational determination of parameter values of anthropometric segments. J. Biomechanics 13, 833-843.

Huston, R. L., and Passerello, C. E. (1971). On dynamics of a human body model. J. Biomech. 4, 369-378.

Jensen, R. K. (1978). Estimation of the biomechanical properties of three body types using a photogrammetric method. J. Biomechanics 11, 349-358.

Kwon, Y.-H. (1996). Effects of the method of body segment parameter estimation on airborne angular momentum. Journal of Applied Biomechanics 12, 413-430.

Kwon, Y.-H. (1994). KWON3D Motion Analysis Package 2.1 User's Reference Manual. Anyang, Korea: V-TEK Corporation.

Kwon, Y.-H. (1993). The effects of body segment parameter estimation on the experimental simulation of a complex airborne movement. Doctoral Dissertation, Pennsylvania State University.

Miller, D. I. and Morrison, W. (1975). Prediction of segmental parameters using the Hanavan human body model. Med. Sci. Sports 7, 207-212.

Rodrigue, D. and Gagnon, M. (1984). Validation of Weinbach's and Hanavan's models for computation of physical properties of the forearm. Res. Q. Exercise Sport 55, 272-277.

Sady, S., Freedson, P., Katch, V. L. and Reynolds, H. M. (1978). Anthropometric model of total body volume for males of different sizes. Human Biol. 50, 529-540.

Weinbach, A. P. (1938). Contour maps, center of gravity, moment of inertia and surface area of the human body. Human Biol. 10, 356-371.

Yeadon, M. R. (1990). The simulation of aerial movement-II. A mathematical inertia model of the human body. J. Biomechanics 23, 67-74.

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© Young-Hoo Kwon, 1998-