page 23

5	5	3

3

10°

5	5	3

3

4 10°

2.3.3 Geometry

****************** ADD IN MASS MOMENTS AND DESCRIPTIONS ************

• A set of the basic 2D and 3D geometric primitives are given, and the notation used is described below,

page 24

A =					contained area
P =					perimeter distance
V = contained volume
S =					surface are
x,y ,z = centre of mass
		,		,z = centroid
	x		y
Ix,Iy ,Iz = moment of inertia of area (or second moment of inertia)
AREA PROPERTIES:
					Ix = ∫y2dA =	the moment of inertia about the y-axis
					A
					Iy = ∫x2dA =	the moment of inertia about the x-axis
					A

	Ixy			=	∫xydA =			the product of inertia
					A
	JO = ∫r2dA =							∫(x2 + y2 )dA = Ix + Iy = The polar moment of inertia
					A			A
					∫
					xdA
				=	A	=	centroid location along the x-axis
	x
					------------
					∫ dA
					A
					∫
					ydA
				=	A	=	centroid location along the y-axis
	y
					------------
					∫ dA

A

page 25

Rectangle/Square:

A = ab

P = 2a + 2b

Centroid:

b x = --2

a y = -- 2

Triangle:

bh A = -----

2

P =

y

a

x

											b
	Moment of Inertia							Moment of Inertia
	(about centroid axes):							(about origin axes):
					=		ba3			=	ba3
	Ix							Ix
							--------				--------
							12				3
					=		b3a			=	b3a
	I							I
			y				--------		y		--------
							12				3
							= 0			=	b2a2
		Ixy						Ixy
											----------
											4

y

a

θ	h
							x
	b

Centroid:

Moment of Inertia

Moment of Inertia

a + b

(about centroid axes):

(about origin axes):

=

x

3

-----------

bh

bh3

3

=

Ix

I

=

--------

x

--------

h

36

12

y

=

3--

=

bh

(a

2

+ b

2

– ab )

bh

(a

2

2

– ab )

Iy

Iy

=

+ b

-----

-----

36

12

			bh2		bh2
	Ixy			Ixy
			= --------(2a – b )		= --------(2a – b )
			72		24

page 26

Circle:

A = π r2

P = 2π r

y

r

x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes): (about origin axes):

π r4

x = r

Ix =

I

=

-------

x

4

π r4

y = r

I

y

=

-------

Iy

=

4

= 0

Ixy

Ixy

=

Half Circle:	y
π r2
A = -------
2
P = π r + 2r	r
	x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

= r

π

8

4

4

x

=

π r

Ix

-- – -----

r

Ix

=

8

9π

-------

4r

4

8

y

= -----

π r

π r

4

3π

Iy

=

-------

I

y

=

-------

8

8

Ixy

= 0

Ixy

= 0

page 27

Quarter Circle:		y
A =	π r2
	-------
	4
P =	π r
	----- + 2r	r
	2
		x

Centroid:

Moment of Inertia

Moment of Inertia

4r

(about centroid axes):

(about origin axes):

π r

4

x

=

-----

4

Ix

=

Ix = 0.05488r

3π

-------

16

4r

4

π r

4

y

=

-----

Iy = 0.05488r

Iy

=

-------

3π

16

4

r4

Ixy = –0.01647r

I

=

xy

----

8

Circular Arc:			y
A =	θ r	2
	-------
	2
P = θ r + 2r			r
			x
			θ

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

θ

4

2r sin --

r

=

2

I

x

=

I

x

= ----(θ – sinθ

)

x

----------------

8

3

θ

r

4

= 0

Iy

=

Iy

= ----(θ + sinθ

)

y

8

=

Ixy = 0

Ixy

page 28

Ellipse:								y	r1
A = π	r1r2								r2
		π	2		2
		--	r1	+ r	2		2
		2				(sin θ			x
P = 4r1∫			1 – -------------------				) dθ
		0	a
P ≈2π	r	12 + r22
	---------------
		2

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes): (about origin axes):

= r

π r13r2

x

Ix =

2

Ix = ------------

4

= r

π r1r23

I

=

y

1

Iy =

------------

y

4

Ixy

=

Ixy

=

Half Ellipse:							y
A =	π r1r2
	------------
	2
	π	2	2				r1
	--	r1 + r	2		2		r2
	2			(sin θ		+ 2r2		x
P = 2r1∫		1 – -------------------			) dθ
	0	a

P ≈π	r	12 + r22
	--------------- + 2r2
		2

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

= r

π r2r13

x

2

3

I

=

------------

Ix = 0.05488r2r1

x

16

=

4r1

3

π r23r1

y

=

0.05488r

=

I

r

I

-------

2

1

------------

3

π

y

y

16

=

–0.01647r

2

2

=

r12r22

I

r

I

xy

1

2

xy

---------

8

page 29

Quarter Ellipse:							y
A =	π r1r2
	------------
	4
	π	2	2				r1
	--	r1 + r	2		2		r2
P =	2			(sin θ		+ 2r2		x
	r1∫	1 – -------------------			) dθ
	0	a

π	r	12 + r22
P ≈--	--------------- + 2r2
2		2

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

4r2

=

=

3

=

Ix

Ix

x

-------

π r2r1

3π

4r1

3

=

Iy

=

Iy

=

π r2r1

y

-------

3π

r22r12

=

=

Ixy

Ixy

---------

8

Parabola:

y

A =

2

--ab

3

a

2

2

2

2

2

b

+ 16a

b

+ 16a

P =

+

4a + b

x

---------------------------

-----

ln ---------------------------------------

2

8a

b

b

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

b

=

=

I

I

x

=

--

x

x

2

=

I

=

=

2a

I

y

y

y

-----

5

=

Ixy

=

Ixy

page 30

Half Parabola:
A =	ab
	-----
	3
	b	2	+ 16a	2		b	2			2	+ 16a	2
P =					+			4a + b
	---------------------------					--------		ln ---------------------------------------
			4			16a			b

y

a

x

b

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes): (about origin axes):

3b

=

8ba3

=

2ba3

=

Ix

Ix

x

-----

-----------175

-----------7

8

19b3a

2b3a

=

=

2a

Iy

Iy

=

--------------480

-----------15

y

-----

5

b2a2

b2a2

=

=

Ixy

Ixy

----------60

----------6

• A general class of geometries are conics. This for is shown below, and can be used to represent many of the simple shapes represented by a polynomial.

Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0

Conditions	A = B = C = 0
	B = 0,A = C
	B2	– AC <0
	B2	– AC = 0
	B2	– AC >0

straight line

circle

ellipse

parabola

hyperbola

page 31

VOLUME PROPERTIES:

Ix = ∫rx2dV = the moment of inertia about the x-axis

V

Iy = ∫ry2dV = the moment of inertia about the y-axis

V

Iz = ∫rz2dV = the moment of inertia about the z-axis

V

				∫
				xdV
			=	V	=	centroid location along the x-axis
	x
				------------
				∫ dV
				V
				∫
				ydV
			=	V	=	centroid location along the y-axis
	y
				------------
				∫ dV
				V
				∫
				zdV
	z		=	V	=	centroid location along the z-axis
				------------
				∫ dV
				V

page 32

Parallelepiped (box):		y
V = abc		c
		z
S = 2(ab + ac + bc )
	b	x
		a

Sphere:

4π 3 V = -- r

3

S = 4π r2

Centroid:

			a
x		=	2--
			b
y		=	2--
			c
z		=	2--

Moment of Inertia (about centroid axes):

				M(a2	+ b2 )
	Ix
				= --------------------------
				12
				M(a2	+ c2 )
	Iy
				= --------------------------
				12
				M(b2	+ a2 )
	Iz
				= --------------------------
				12

y

Moment of Inertia (about origin axes):

Ix =

Iy =

Iz =

r

z

x

	Centroid:				Moment of Inertia					Moment of Inertia
					(about centroid axes):					(about origin axes):
									2Mr2
				= r	Ix			=	------------	Ix	=
	x
									5
									2Mr2
				= r	Iy			=	------------	Iy	=
	y
									5
								=	2Mr2		=
					Iz					Iz
	z			= r					------------
									5

page 33

Hemisphere:			y
2	π	r	3
V = --
3			z
S =			r	x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

83

2

x

= r

Ix

= --------Mr

Ix =

320

2Mr2

3r

Iy

=

------------

Iy

=

y

= ----

5

8

83

=

2

=

Iz

Iz

z

= r

--------Mr

320

Cap of a Sphere:				y
1		2	(3r – h )	h
V = --π h
3				z
S = 2π
	rh			r
				x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

= r

Ix

x

=

Iy

=

=

Iy

y

=

Iz

=

z

= r

Iz

page 34

Cylinder:		y	r
V = hπ r2			h
S = 2π rh + 2π r2			z
			x
Centroid:	Moment of Inertia		Moment of Inertia
	(about centroid axis):		(about origin axis):

=

h2

r2

h2

r2

x

r

Ix

=

Ix

=

M -----

+ ----

M ----

+ ----

12

4

3

4

h

Mr2

y

=

Iy

=

---------

Iy

=

--

2

2

=

h2

r2

=

h2

r2

I

I

M

z = r

M -----

+ ----

----

+ ----

z

12

4

z

3

4

Cone:						y
V =	1	π	r	2	h
	--
	3
S = π r r2 + h2						z	h
							r
							x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

3h3

3r2

x =

r

Ix

=

Ix

=

M --------

+ -------

80

20

h

=

3Mr2

=

=

Iy

Iy

y

--

------------

4

10

=

3h3

3r2

=

I

I

z

= r

M --------

+ -------

z

80

20

z

page 35

Tetrahedron:		y
1		z
V = --Ah
3		h
		A	x
Centroid:	Moment of Inertia	Moment of Inertia
	(about centroid axes):	(about origin axes):

			=		Ix				=	Ix	=
	x
									=	Iy	=
				h	Iy
y			=	4--
									=	Iz	=
z			=			Iz

Torus:					y
1					r2
	π	2	(r1	+ r2 )(r2	2
V = --					– r1 )
4					r	1
					z
S = π 2(r22 – r12 )						x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

= r2

Ix

x

r2

– r1

=

I

=

I

y

y

y

= --------------

2

=

Iz

=

z

= r2

Iz

page 36

Ellipsoid:	y
4	r2	r3
	π r1r2r3
V = --
3	z	r1
S =		x

	Centroid:				Moment of Inertia					Moment of Inertia
					(about centroid axes):					(about origin axes):
									=	Ix	=
				= r1		Ix
	x
									=	Iy	=
				= r2		Iy
	y
									=	Iz	=
	z			= r3			Iz

Paraboloid:					y
V =	1	π r	2	h
	--				h
	2
					z
S =					r	x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

= r

Ix

x

=

Iy

=

=

Iy

y

=

Iz

=

z

= r

Iz