Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
1. The pressure angle is the angle between the direction of motion (velocity) of the follower and
the direction of the axis of transmission.
a) When becomes 90 there will be no motion of the follower.
In this figure 8.1 (a) on Page 403, velocity of follow
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
GAME THEORY
Life is full of conflict and competition.
Numerical examples involving adversaries in
conflict include parlor games, military battles,
political campaigns, advertising and
marketing campaigns by competing business
firms and so forth. A basic f
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
(12/ L = d1 /24) hence d1 = 9 and
average of 9 and 12 is 10.5in
Potential Energy Minimization Method
External force components
f body force acting on volume;
T traction force acting on surface and
P point force acting at a point
Strain energy is related t
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
QUADRIC POLYNOMIAL FUNCTIONS
Represents a conic section curve
x2 + y2 1 = 0
a 2 b2
Ellipse
Parabola
y 2 4 a x= 0
Hyperbola
x2 y2 1 = 0
2
a2
b
2
2
circle x y r
Ellipse
2
Hyperbola
Parabola
Circle
1
Parabola
It is a curve generated by a point that moves
su
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
SOLID MODELLING
Why solid modeling?
Recall weakness of wireframe and surface
modeling
Ambiguous geometric description
incomplete geometric description
lack topological information
Tedious modeling process
Awkward user interface
Solid model
Solid mo
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Rapid Prototyping
Definition
Rapid Prototyping (RP) can be defined as
a group of techniques used to quickly
fabricate a scale model of a part or
assembly using threedimensional
computer aided design (CAD) data.
Why Rapid Prototyping?
The reasons of Rapi
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
The curves do not meet at all.
C0 The endpoints of the two
curves meet (the curves have
positional continuity only).
There may be a sharp point
where they meet.
C1The curves have identical
tangents at the breakpoint. (The
tangent is the slope at the
bre
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Beam Element
Translation due to force and moment for i,
Angular deflection at i due to force and moment,
The matrix can be used directly for this class of problems.
Note that, the matrix will be different for different beams
For axial loading on element 3
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Finite Element Equation of a Truss Structure
In this section, we will derive the finite element equation of a truss structure. The
procedure presented here is the basis for all FEA analyses formulations, wherever helement are used.
Analogues to the previo
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Two dimensional stress analysis
For 3D
G = E/2(1 + v)
Solve the equations to determine the displacements.
In matrix form,
Maths .
For three general vertices, the equation for area of triangle is:
Ni Ui
Ni vi
Shape functions
Internal energy  External work
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Direct stiffness approach
Taking inverse of stiffness matrix and multiplying it with force vector
will give you the displacement vector.
Practice matrix inversion and gauss elimination method for solution.
Review:
1.
P
2. Explain the basic procedure for F
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Foreshortening refers to the visual effect or
optical illusion that an object or distance
appears shorter than actual
Oblique projection
If d is greater than 0
To get the positions of x and y in Cartesian coordinates, divide (1z/d) from
the Pv vector.
Pe
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
Uses of transformation:
Y axis
X axis
To shear in the x direction the equation is:
x1 = x + ay
y1 = y
Where b = 0
Where x1 and y1 are the new values, x and y are the original values,
and a is the scaling factor in the x direction. The matrix is as follows
Birla Institute of Technology & Science, Pilani  Dubai
CAD
MECHANICAL ME F344

Spring 2016
CAD/CAM/CAE
ComputerAided Design (CAD) is the technology concerned with
the use of computer systems to assist in the creation, modification,
analysis, and optimization of a design.
ComputerAided Manufacturing (CAM) is the technology concerned
with the
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Deterministic Dynamic Programming
Dynamic Programming (DP) determines the
optimum solution to an nvariable problem by
decomposing it into n stages with each stage
constituting a singlevariable sub problem.
Recursive Nature of Computations in DP
Computat
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
PERT Networks
In PERT the duration of any activity is
indeterministic. It bases the duration of an
activity on three estimates:
Optimistic Time, a
Most Likely Time, m
Pessimistic Time, b
The range [a, b] is assumed to enclose all
possible estimates of
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Algebraic Solution of LPPs  Simplex
Method
To solve an LPP algebraically, we first put it
in the standard form. This means all
decision variables are nonnegative and all
constraints (other than the nonnegativity
restrictions) are equations with nonnegati
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
In this presentation
we illustrate the ideas
developed in the
previous presentation
with two more
problems
Consider the following LPP:
Maximize z 6 x1 x2 2 x3
Subject to
1
2 x1 2 x2 x3 2
2
3
4 x1 2 x2 x3 3
2
1
x1 2 x2 x3 1
2
x1 , x2 , x3 0
Let x4, x5, x6
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Explanation of the
entries in any simplex
tableau in terms of the
entries of the starting
tableau
In this lecture we explain how the
starting Simplex tableau (in matrix form)
gets transformed after some iterations.
We also give the meaning of the entries
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
MATRIX FORMULATION
OF THE LPps
In this lecture we shall look at the matrix
formulation of the LPPs. We see that the
Basic feasible solutions are got by solving
the matrix equation BX b
where B is a mm nonsingular submatrix
of the contraint matrix of the L
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Duality theorems
Finding the dual optimal
solution from the primal
optimal tableau
Dual problem in Matrix form
In this lecture we shall present the primal
and dual problems in matrix form and
prove certain results on the feasible and
optimal solutions of
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Dual Problem of an LPP
Given a LPP (called the primal problem),
we shall associate another LPP called the dual
problem of the original (primal) problem. We
shall see that the Optimal values of the primal
and dual are the same provided both have
finite fea
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
In this presentation
we illustrate the ideas
developed in the
previous presentation
with two more
problems
Consider the following LPP:
Maximize z 6 x1 x2 2 x3
Subject to
1
2 x1 2 x2 x3 2
2
3
4 x1 2 x2 x3 3
2
1
x1 2 x2 x3 1
2
x1 , x2 , x3 0
Let x4, x5, x6
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Some problems
illustrating the principles
of duality
In this lecture we look at some
problems that uses the results
from Duality theory (as discussed
in Chapter 7).
Problem 7. Problem Set 4.2D Page 130
Consider the LPP
Maximize z 5 x1 2 x2 3 x3
subject to
Birla Institute of Technology & Science, Pilani  Dubai
Engineering Optimization
MECHANICAL ME F344

Summer 2015
Sensitivity Analysis
The optimal solution of a LPP is based on the
conditions that prevailed at the time the LP model
was formulated and solved. In the real world, the
decision environment rarely remains static and it
is essential to determine how the opt