chapter 9_centroid and cg

7/28/2019 Chapter 9_Centroid and CG

1/24

Chapter 9Center of Gravity and

Centroids (9.2 only)

Note, center of gravity location =centroid location for homogenous

material (density or specific weight isconstant throughout the body).


2/24

APPLICATIONS (continued)

One of the important factors in determining its stability is theSUVs center of mass.

Should it be higher or lower to make a SUV more stable?

How do you determine the location of the SUVs center of mass?

One concern about a sport utility vehicle (SUV) is that it might tip

over while taking a sharp turn.


3/24

APPLICATIONS (continued)

Integration must be used to

determine total weight of the goal

post due to the curvature of the

supporting member.

How do you determine the

location of its center of

gravity?

To design the ground supportstructure for the goal post, it is

critical to find total weight of the

structure and the center of gravity

location.


4/24

CONCEPT OF CENTER OF GRAVITY (CG)

A body is composed of an infinite number of

particles, and so if the body is located within a

gravitational field, then each of these particles

will have a weight dW.

From the definition of a resultant force, the sum

of moments due to individual particle weighst

about any point is the same as the moment due

to the resultant weight located at G.

Also, note that the sum of moments due to the individual particles

weights about point G is equal to zero.

The center of gravity (CG) is a point, often

shown as G, which locates the resultant weight

of a system of particles or a solid body.


5/24

CONCEPT OF CG (continued)

The location of the center of gravity, measured

from the y axis, is determined by equating themoment of W about the y axis to the sum of the

moments of the weights of the particles about this

same axis.

~ ~ ~

x W =x dW~_IfdWis located at point (x, y, z), then

Similarly, y W =y dW~_

z W =z dW~_

Therefore, the location of the center of gravity G with respect to the x,y,z axes becomes


6/24

CONCEPT OF CENTROID

The centroid coincides with the center of mass or the center of gravity only if

the material of the body is homogenous (density or specific weight is constant

throughout the body).

If an object has an axis of symmetry, then the centroid of object lies on that

axis.

In some cases, the centroid is not located on the object (u-shape).

The centroid, C, is a point which defines the geometric center of an object.

x = ( A x dA ) / ( A dA )~

y = ( A y dA ) / ( A dA )~

Equations:


7/24

Centroids of Common Areas

Note: Centroid = CG for homogenous

body!!!


8/24

Centroids of Common Areas


9/24

How can we easily determine

the location of the centroid for

a given beam shape?

The I-beam or tee-beamshown are commonly used in

building various types of

structures.

When doing a stress ordeflection analysis for a beam,

the location of the centroid is

very important.

COMPOSITE BODIES


10/24

CONCEPT OF A COMPOSITE BODY

Knowing the location of the centroid, C, or center of gravity, G,

of the simple shaped parts, we can easily determine the location

of the C or G for the more complex composite body.

Many industrial objects can be considered as composite bodies

made up of a series of connected simple shaped parts or holes,

like a rectangle, triangle, and semicircle.


11/24

CONCEPT OF A COMPOSITE BODY

(continued)

This can be done by considering each part as a particle and

following the procedure as described in Section 9.1.

This is a simple, effective, and practical method of determining

the location of the centroid or center of gravity of a complex

part, structure or machine.


12/24

STEPS FOR ANALYSIS

1. Divide the body into pieces that are known shapes.

Holes are considered as pieces with negative weight or size.

2. Make a table with the first column for segment number, the second

column for weight, mass, or size (depending on the problem), the

next set of columns for the moment arms, and, finally, several

columns for recording results of simple intermediate calculations.

3. Fix the coordinate axes, determine the coordinates of the center of

gravity of centroid of each piece, and then fill in the table.

4. Sum the columns to get x, y, and z. Use formulas like

x = ( xi Ai ) / ( Ai ) or x = ( xi Wi ) / ( Wi )

This approach will become clear by doing examples!


13/24

EXAMPLE

Solution:

1. This body can be divided into the following pieces:

rectangle (a) + triangle (b) + quarter circular (c)semicircular area (d). Note the negative sign on the hole!

Given: The part shown.

Find: The centroid of

the part.

Plan: Follow the steps

for analysis.


14/24

EXAMPLE (continued)

39.8376.528.0

27

4.5

9

- 2/3

54

31.5

9

0

1.5

1

4(3) / (3 )4(1) / (3 )

3

7

4(3) / (3 )0

18

4.5

9 / 4 / 2

Rectangle

Triangle

Q. CircleSemi-Circle

y A( in3)

x A( in3)

y(in)

x(in)

Area A(in2)

Segment

Steps 2 & 3: Make up and fill thetable using parts a, b,

c, and d.


15/24

4. Now use the table data results and the formulas to find thecoordinates of the centroid.

EXAMPLE

(continued)

x = ( x A) / ( A ) = 76.5 in3/ 28.0 in2 = 2.73 in

y = ( y A) / (A ) = 39.83 in3 / 28.0 in2 = 1.42 in

C

Area A x A y A

28.0 76.5 39.83


16/24

CONCEPT QUIZ

1. Based on the typical centroid

information, what are the minimumnumber of pieces you will have to

consider for determining the centroid of

the area shown at the right?

A) 1 B) 2 C) 3 D) 4

2. A storage box is tilted up to clean the rug

underneath the box. It is tilted up by pulling

the handle C, with edge A remaining on the

ground. What is the maximum angle of tilt(measured between bottom AB and the

ground) possible before the box tips over?

A) 30 B) 45 C) 60 D) 90

3cm 1 cm

1 cm

3cm

30

G

C

AB


17/24

Examples: Find Centroids for these cross-sections:

Side, CG?


18/24

Centroid of Composite Areas

Missing Areas canbe calculated into

the Centroid byanalyzing it as anegative area.


19/24

Sample Problem - Hole


20/24

Sample Problem - Hole

23

33

mm1013.828

mm107.757

A

AxX

mm8.54X

23

33

mm1013.828

mm102.506

A

AyY

mm6.36Y

Compute the coordinates of the area

centroid by dividing the first moments by

the total area.


21/24

CG / CM OF A COMPOSITE BODY

By replacing the W with a M in these equations, the coordinates

of the center of mass can be found.

Summing the moments about the y-axis, we get

x WR = x1W1 + x2W2+ .. + xnWnwhere x1 represents x coordinate of W1, etc..

~~~

~

Consider a composite body which consists of a

series of particles(or bodies) as shown in the

figure. The net or the resultant weight is given

as WR= W.

Similarly, we can sum moments about the x- and z-axes to find

the coordinates of G.


22/24

Example: Center of Gravity, 3D with different densities:

Solution

1. In this problem, the blocks A and B can be considered as two

pieces (or segments).

Given: Two blocks of different

materials are assembled asshown.

The densities of the materials are:

A

= 150 lb / ft3 and

B = 400 lb / ft3.

Find: The center of gravity of this

assembly.

Plan: Follow the steps for analysis.


23/24

GROUP PROBLEM SOLVING (continued)

56.2553.1229.1719.79

6.25

50.00

3.125

50.00

12.5

16.67

2

3

1

3

4

1

3.125

16.67

A

B

zA

(lbin)

yA

(lbin)

xA

(lbin)

z (in)y (in)x (in)w (lb)Segment

Weight = w = (Volume in ft3)

wA = 150 (0.5) (6) (6) (2) / (12)3 = 3.125 lb

wB = 400 (6) (6) (2) / (12)3 = 16.67 lb


24/24

GROUP PROBLEM SOLVING (continued)

~x = ( x w) / ( w ) = 29.17/19.79 = 1.47 in

y = ( y w) / ( w ) = 53.12/ 19.79 = 2.68 in

z = ( z w) / ( w ) = 56.25 / 19.79 = 2.84 in~

~

W (lb) x w y w z w(lbin) (lbin) (lbin)

19.79 29.17 53.12 56.25

Table Summary

Substituting into the equations:

chapter 9_centroid and cg

Documents