ES 13: Mechanics of Deformable Bodies I
Lecture No. 19
Beam Deflections using
Area-Moment Method (AMM)
Euler-Bernoulli Equation
Moment-Area Theorems
Area-Moment Method (AMM)
Moment Diagram by-Parts
Maximum Beam Deflection
Jessica M. Junio
Institute of Civ
ES 13: Mechanics of Deformable Bodies I
Lecture No. 18
Beam Deflections using Double
Integration Method (DIM)
Governing Equation for Euler
Bernoulli
Macaulay Functions
Double Integration Method (DIM)
Jessica M. Junio
Institute of Civil Engineering
Univ
Analytic
Geometry
MATH 36 A
MATHEMATICAL ANALYSIS I
Lecture 15
18 March 2015
Limits &
Continuity
Derivatives
Applications of
Derivatives
Integration
1
CHAPTER 3
THE DERIVATIVE AND DIFFERENTIATION
Outline
1. Definition of the Derivative and its
Geometric R
Unit 3
Applications of the
Definite Integral
Unit 3.1
Volume of Solids of
Revolution
Unit 3.1.1
Volume of Solids of
Revolution
(Using Disks)
Solid of Revolution
A solid of revolution is a solid obtained by
revolving a region in a plane about a line in
the
Print this file using A4 and write your solutions on it.
EXERCISE 10
Center of Mass
Week 13 MATH 37 X
Due on November 3, 2016 in the lecture class
NAME:_ RECIT SECTION:_ SCORE:_
SET-UP only. All or nothing per set-up.
1. A 3-ft long rod has a density that
Unit 5.5
Equations of Lines
and Planes
Plane in Space
A plane in space is determined by a point
P x0 , y0 , z0 in the plane and a vector N that
is orthogonal to the plane.
This orthogonal vector N is called a normal
vector of the plane.
N
P x0 , y0 , z0
Unit 3.1.2
Volume of Solids of
Revolution
(Using Cylindrical Shells)
Consider the region that lies under the curve
y f x , above the x-axis and between the
lines x a and x b.
Subdivide the region into n strips of equal width.
xi 1
xi
Each strip has width
Unit 5.3
Dot Product
Dot Product
If A a1, a2 and B b1, b2 , then the dot
product of A and B is given by
A B a1b1 a2b2
If A a1, a2 , a3 and B b1, b2 , b3 , then
A B a1b1 a2b2 a3b3
Example 5.3.1
Perform the indicated operation.
1.
2,1,3 1,0, 5 2 1 1 0 3 5
Unit 3.4.2
Center of Mass
of a Plane Region
n
x
m x
i 1
n
i i
m
i 1
i
mi xi moment of mass of the ith particle
with respect to the origin
n
M 0 mi xi moment of mass of the system
i 1
with respect to the origin
n
M mi total mass of the system
i 1
Center of
4.3
AREA
of
POLAR
REGIONS
1
Area of a sector of a circle.
1 2
A r
2
Consider the shaded region below.
Note: It is assumed here that r f 0.
Subdivide , into n subintervals of equal length.
Intervals: 0 ,1 ,1,2 ,.,i 1,i ,.,n 1,n
Choose i *i 1,i
2
Unit 5
Vectors and the Space
1
Unit 5.1
The Three Dimensional
Coordinate System
2
xy plane:
z0
xz plane:
y0
yz plane:
x0
P x, y , z
z
y
x
Plotting Points in 3D
For P x, y, z ,
1. Locate the point x, y on the xy plane.
2. If z is positive, locate the poin
Unit 4.2
Graphs of Polar Equations
Graph of a Polar Equation
The graph of a polar equation F r, 0,
consists of all points r , whose coordinates
satisfy the equation.
Lines
Form: C
Graph: line passing through the pole and
making angle of measure C radians
Unit 5.2
Vectors
Vector
A vector is used to indicate a quantity that has
both magnitude and direction.
Velocity indicates speed magnitude and
direction
positive/negative .
Q
P
CD
AB
Vector in Plane and Space
A vector in
2
is an ordered pair of real numbe
Unit 3.3
Area of a Surface of
Revolution
Surface of Revolution
A surface of revolution is formed when a curve
is revolved about a line.
SA 2 rh
SA r1 r2 l
Consider the surface obtained by revolving the
curve y f x , a x b, about the x axis.
Area of a Surf
Unit 3.4.3
Center of Mass
of a Solid of Revolution
xy plane:
xz plane:
yz plane:
z0
y0
x0
Consider a system of n particles with masses
m1, m2 , , mn located at the points x1, y1, z1 ,
x2 , y2 , z2 , , xn , yn , zn in the space.
The center of mass of the
Unit 4
Polar Coordinate System
Unit 4.1
Cartesian and Polar
Coordinate System
Polar Coordinate System
A polar coordinate system consists of a horizontal
ray called the polar axis.
The initial point of the polar axis is called the pole.
pole
polar axis
Pol
4/10/2015
MATH 36 A
Analytic
Geometry
MATHEMATICAL ANALYSIS I
Lecture 16
27 March 2015
Limits &
Continuity
Derivatives
Applications of
Derivatives
Integration
1
CHAPTER 3
THE DERIVATIVE AND DIFFERENTIATION
Outline
1. Definition of the Derivative and its
G
4/10/2015
MATH 36 A
Analytic
Geometry
MATHEMATICAL ANALYSIS I
Lecture 17
08 April 2015
Limits &
Continuity
Derivatives
Applications of
Derivatives
Integration
1
CHAPTER 3
THE DERIVATIVE AND DIFFERENTIATION
Outline
1. Definition of the Derivative and its
G
Unit 5.6
Cylinders and
Quadric Surfaces
Traces
A trace of a surface is a curve of intersection
of the surface with planes parallel to the
coordinate planes.
Cylinders
A cylinder is a surface that consists of all lines
(rullings) that are parallel to a giv
Unit 3.2
Length of an Arc
Consider the graph of y f x where f is
continuous and a x b.
y f x
a
b
Length of an Arc
If f ' is continuous on a, b , then the length
of the curve y f x , a x b is
L
b
a
1 f ' x dx.
2
Similarly, if a curve has the equation x g y
3/18/2015
MATH 36 A
Analytic
Geometry
MATHEMATICAL ANALYSIS I
Lecture 14
18 March 2015
Limits &
Continuity
Derivatives
Applications of
Derivatives
Integration
1
CHAPTER 3
THE DERIVATIVE AND DIFFERENTIATION
Outline
1. Definition of the Derivative and its
G
Unit 3.4
Center of Mass
Unit 3.4.1
Center of Mass
of a Rod
The rod will balance if m1d1 m2d2
m1d1 m2d2
m1 x x1 m2 x2 x
m1x m1x1 m2 x2 m2 x
m1x m2 x m1x1 m2 x2
m1x1 m2 x2
x
m1 m2
In general, in a system of n particles with masses
m1, m2 , , mn located at
Unit 5.4
Cross Product
Cross Product
If A a1 , a2 , a3 and B b1 , b2 , b3 , then the
cross product of A and B is the vector
A B a2b3 a3b2 , a3b1 a1b3 , a1b2 a2b1
A B a2b3 a3b2 , a3b1 a1b3 , a1b2 a2b1
i
j
k
A B a1
a2
a3
b1
b2
b3
a2
a3
b2
b3
i
a1
a3
b1
b3
j
ENSC 11 PROBSET 2
2nd Semester 2014-2015
1. For the given truss:
a. Use the Method of Section to find the forces in members FL, PQ, and GQ;
b. Use the Method of Joints to find the forces in members HN, HM, and MS.
Indicate whether the required members are
C H A P T E R 1.1
Principles of
Statics
SAMPLE PROBLEMS
Fundamentals of Statics
Sample Problem 1
Determine the magnitude and direction
(measured clockwise from the x-axis) of the
resultant force acting on the screw eye.
Fundamentals of Statics
Sample Prob
DEPARTMENT OF ENGINEERING SCIENCE
COLLEGE OF ENGINEERING AND AGRO-INDUSTRIAL TECHNOLOGY
UNIVERSITY OF THE PHILIPPINES LOS BAOS
ENGINEERING SCIENCE 11
STATICS OF RIGID BODIES
PROBLEM SET 1
PROBLEM 1
a. If the resultant force of the two tugboats is 3 kN, di
International Journal for Quality Research 7(1) 127140
ISSN 1800-6450
Ripon Kumar
Chakrabortty1
Tarun Kumar Biswas
Iraj Ahmed
Article info:
Received 15 November 2012
Accepted 26 February 2013
UDC 65.018
REDUCING PROCESS VARIABILITY BY
USING DMAIC MODEL: A
AN INTERACTIVE PROGRAM
TO BALANCE ASSEMBLY LINES
John J. Bartholdi, III
1992; April 3, 2003
Abstract
We describe the development of a program to balance 1- or 2-sided
assembly lines for a manufacturer of utility vehicles. The program is
highly interactive
Weights and Measures Program
Requirements
A Handbook for the
Weights and Measures Administrator
Authors:
Carol Hockert, Chief
Henry V. Oppermann
National Institute of Standards and Technology
Weights and Measures Division
Gaithersburg, MD 20899-2600
U. S.
International Conference on Product Lifecycle Management
1
Line Balancing in the Real World
Emanuel Falkenauer
Optimal Design
Av. Jeanne 19A bote 2, B-1050 Brussels, Belgium
+32 (0)2 646 10 74
[email protected]
Abstract: Line Balancing (LB) i