ME221 Lecture 7 1

ME 221 Statics

Lecture #7

Sections 4.1 – 4.3

ME221 Lecture 7 2

Homework #3• Chapter 3 problems:

– 48, 55, 57, 61, 62, 65 & 72

• Chapter 4 problems:– 2, 4, 10, 11, 18, 24, 39 & 43– Must use integration methods to solve

• Due Monday, June 7

ME221 Lecture 7 3

Quiz #4

• Monday, June 7

ME221 Lecture 7 4

Distributed Forces (Loads);Centroids & Center of Gravity

• The concept of distributed loads will be introduced

• Center of mass will be discussed as an important application of distributed loading

– mass, (hence, weight), is distributed throughout a body; we want to find the “balance” point

ME221 Lecture 7 5

Distributed LoadsTwo types of distributed loads exist:

– forces that exist throughout the body• e. g., gravity acting on mass

• these are called “body forces”

– forces arising from contact between two bodies• these are called “contact forces”

ME221 Lecture 7 6

Contact Distributed Load• Snow on roof, tire on road, bearing on race,

liquid on container wall, ...

ME221 Lecture 7 7

Center of Gravity

x

y

z

w5(x5,y5,z5)˜ ˜ ˜

x

y

z

w3(x3,y3,z3)˜ ˜ ˜w1(x1,y1,z1)˜ ˜ ˜

w2(x2,y2,z2)˜ ˜ ˜ w4(x4,y4,z4)˜ ˜ ˜

The weights of the n particles comprise a system of parallel forces. We can replace them with an equivalent force w located at G(x,y,z), such that:

x w=x1w1+x2w2+x3w3+x4w4 +x5w5~ ~ ~ ~ ~

y

zx

ME221 Lecture 7 8

Or

n

ii

n

iii

n

ii

n

iii

n

ii

n

iii

w

wzz

w

wyy

w

wxx

1

1

1

1

1

1

~

,

~

,

~

Where are the coordinates of each point. Point G is called the center of gravity which is defined as the point in the space where all the weight is concentrated.

zyx ~,~,~

ME221 Lecture 7 9

CG in Discrete Sense

Where do we hold the bar to balance it?

20 10

???? ??

Find the point where the system’s weight may be balanced without the use of a moment.

ME221 Lecture 7 10

Discrete Equations

Define a reference framex

y

z

dwr

dwdwx

xwwx

xi

ii

~~

ME221 Lecture 7 11

. . . . . .; ;i i i i i i

i i ic m c m c m

m x m y m z

x y zM M M

Mass center is defined by

The total mass is given by M

i

i

M m

Center of Mass

ME221 Lecture 7 12

Continuous EquationsTake our volume, dV, to be infinitesimal.

Summing over all volumes becomes an integral.

. . . . . .1 1 1

; ;c m c m c m

V V V

x xdV y ydV z zdVV V V

1

V

M dVV

Note that dm = dV . Center of gravity deals with forces andgdm is used in the integrals.

ME221 Lecture 7 13

If is constant

dvdvz

zdvdvy

ydvdvx

x~

,~

,~

•These coordinates define the geometric center of an object (the centroid)

dAdAz

zdAdAy

ydAdAx

x~

,~

,~

•In case of 2-D, the geometric center can be defined using a differential element dA

ME221 Lecture 7 14

If the geometry of an object takes the form of a line (thin rod or wire), then the centroid may be defined as:

dLdLz

zdLdLy

ydLdLx

x~

,~

,~

ME221 Lecture 7 15

Procedure for Analysis1-Differential element

Specify the coordinate axes and choose an appropriate differential element of integration.

•For a line, the differential element is dl

•For an area, the differential element dA is generally a rectangle having a finite height and differential width.

•For a volume, the element dv is either a circular disk having a finite radius and differential thickness or a shell having a finite length and radius and differential thickness.

ME221 Lecture 7 16

2- SizeExpress the length dl, dA, or dv of the element in terms of the coordinate used to define the object.

3-Moment Arm

Determine the perpendicular distance from the coordinate axes to the centroid of the differential element.

4- EquationSubstitute the data computed above in the appropriate equation.

ME221 Lecture 7 17

x

y

Symmetry Conditions

•In the case where the shape of the object has an axis of symmetry, then the centroid will be located along that line of symmetry.

In this case, the centroid is located along the y-axis

•The centroid of some objects may be partially or completely specified by using the symmetry conditions

ME221 Lecture 7 18

In cases of more than one axis of symmetry, the centroid will be located at the intersection of these axes.

ME221 Lecture 7 19

Centroid of an Area• Geometric center of the area

– Average of the first moment over the entire area

– Where:

A

c xdAA

x1

A

c ydAA

1y

A

dAA

ME221 Lecture 7 20

Centroid of an Area

• Is then defined as an integral over the area.

• Integration of areas may be accomplished by the use of either single integrals or double integrals.

ME221 Lecture 7 21

Centroid of a Volume• Geometric center of the volume

– Average of the first moment over entire volume

– In vector notation:

V

c xdVV

1x

V

c ydVV

1y

V

rdVV

1rc

V

c zdVV

1z

ME221 Lecture 7 22

Examples

ME221 Lecture 7 23

Homework Assignments 4, 5 & 6• Combination of hand-calculated and computer-generated solutions (MatLab). 15

points possible for each.

• Must register for 1 of 2 MatLab sessions on Wednesdays June 9, 16 & 23 (12:40-2:30pm or 5:00-6:50pm).

• ME221 Wednesday lectures will be 10:20am to 11:10am.

• Will be assigned to a group with 2 ME221 & 2 CSE131 students.

• Group members will receive the same grade for MatLab part.