ME 210 Applied Mathematics for Mechanical Engineers
COMPLEX ANALYSIS
Complex Numbers
Complex numbers are useful in the fields such as:
Solutions of some types of linear differential equations
Electrical circuit analyses
Inverse transformations
Solutions o
ME 210 Applied Mathematics for Mechanical Engineers
r
r r
Note that the unit vectors, T , N and B are three unit vectors perpendicular to each
other whose directions are dictated by the local behavior of the curve C at its point P.
They form a moving righ
ME 210 Applied Mathematics for Mechanical Engineers
ME 210
APPLIED MATHEMATICS FOR
MECHANICAL ENGINEERS
Sections 02 & 03
http:/www.me.metu.edu.tr/courses/me210
Prof. Dr. Blent E. Platin
Prof. Dr. Blent E. Platin
Spring 2014 Sections 02 & 03
1/20
ME 210 Ap
ME 210 Applied Mathematics for Mechanical Engineers
Scalar and Vector Fields
Scalar: A geometrical or physical quantity that can completely be characterized by
a single number.
For example: length of a bar, mass of an object, electrical resistivity of a
ME 210 Applied Mathematics for Mechanical Engineers
Theorem: Consider two matrices [A] and [B] of the same size n with n distinct
eigenvalues; if their eigenvalues are the same set then these two matrices are
similar to each other, if [A] and [B] do not h
MIDDLE EAST TECHNICAL UNIVERSITY
DEPARTMENT OF MECHANICAL ENGINEERING
ME 304 Control Systems
Spring 2016
Prepared by: Tayfun Efe Ertop
Abdlhamit Dnder
Sedat Pala
Submission place: G-134
HOMEWORK 02 Solution
Assigned: 25.03.2016
Due date: 01.04.2016
Due ti
MIDDLE EAST TECHNICAL UNIVERSITY
DEPARTMENT OF MECHANICAL ENGINEERING
ME 304 Control Systems
Spring 2016
Prepared by: Tayfun Efe Ertop
Sedat Pala
Abdlhamit Dnder
Submission place: G-134
HOMEWORK 01 Solution
Assigned: 08.01.2016
Due date: 15.01.2016
Due ti
ME 302 THEORY OF MACHINES II
SPRING 2015-2016
HOMEWORK #2 SOLUTION
Prepared by
: Musab ar Uurlu (D-105)
Assignment Date
: March 14, 2016
Due Date
: March 21, 2016 (submit to D-105 until 17:00)
Problem 1:
A slender rod AB, of mass and mass moment of
inerti
ME 302 THEORY OF MACHINES II
HOMEWORK 3
Prepared by Musab ar Uurlu
Room
: D-105
E-mail
: [email protected]
Assignment Date : 22.03.2017
Due date
: 29.03.2017 12:00
Submit your homework to D-105
Problem Three machines are driven by pulley-belt systems from
MIDDLE EAST TECHNICAL UNIVERSITY
DEPARTMENT OF MECHANICAL ENGINEERING
ME 210 APPLIED MATHEMATICS FOR MECHANICAL ENGINEERS
SPRING 2016
HOMEWORK 4
Prepared by: Berke Harmanc, A-145
Submit the solutions of Problem 1 and Problem 2 until 17:00 on April 20, 201
ME 210 Applied Mathematics for Mechanical Engineers
Example: (Path Independence)
r
r
r
4
3
2 2
Evaluate the work done by a force F(x, y) = (10x 2xy ) i 3x y j when its
point of application moves along
a) a closed curve C starting from a point and ending a
ME 210 Applied Mathematics for Mechanical Engineers
Example:
For a mechanical part with a shape of one half turn of a circular helix given by the
r
r r
r
position vector r (t) = cost i + sint j + tk from (1,0,0) to (1,0,), whose mass density
z
per unit le
ME 210 Applied Mathematics for Mechanical Engineers
Limits, Continuity and Derivatives of Complex Functions
Limit: Assume that a complex function w = f(z) is defined in a domain D in the
z-plane as shown in the Figure:
y
v
zplane
w = f(z)
mapping
zoz
wpl
ME 210 Applied Mathematics for Mechanical Engineers
Properties of Determinants:
1. The value of a determinant is zero if it has at least one row (column) of zeros
Examples:
3 1 3
D1 = 1 0 1 = 0
0 0 0
3 0 3
D2 = 1 0 1 = 0
7 0 4
2. The value of a determinan
ME 210 Applied Mathematics for Mechanical Engineers
Eigenvalues and Eigenvectors
The matrix-vector equation
[y] = [A] [x]
where [y] and [x] are n1 column vectors, and [A] is an nn square matrix, can be
considered as a transformation of the vector [x] into
ME 210 Applied Mathematics for Mechanical Engineers
Tangent and Arc Length of a Curve
The tangent to a curve C at a point A on it is defined as the limiting position of the
straight line L through A and B, as B approaches A along the curve as illustrated
ME 210 Applied Mathematics for Mechanical Engineers
Gauss(ian) Elimination
This is a systematic method in which
First variable is eliminated from the (m1) equations starting from the 2nd
equation up until the last equation using 1st equation,
Second varia
ME 210 Applied Mathematics for Mechanical Engineers
INTRODUCTION TO LINEAR ALGEBRA
Matrices and Vectors
Prof. Dr. Blent E. Platin
Spring 2014 Sections 02 & 03
1/23
ME 210 Applied Mathematics for Mechanical Engineers
INTRODUCTION TO LINEAR ALGEBRA
Matrices
ME 210 Applied Mathematics for Mechanical Engineers
r 2r
r
The vector function r (t) = t i + t j may be interpreted as the position
Example:
vectors of points with coordinates P(x,y) where x = t and y = t2 .
Such points are the points
on the parabola y =
ME 210 Applied Mathematics for Mechanical Engineers
Evaluation of surface integrals for a special case
If the surface S of integration is defined as z = g(x,y) then one of its possible
parametric representations is obtained by letting u = x and v = y as
r
ME 210 Applied Mathematics for Mechanical Engineers
Divergence of a Vector Function
r
Let v(x, y, z) be a differentiable vector function with respective Cartesian
components v1, v2 and v3
(
)
r
r
r v v
r
r r r r
v 3
1
2
div v = v =
i+
j + k v1 i + v 2 j
ME 302 THEORY OF MACHINES II
SPRING 2015-2016
HOMEWORK #1 SOLUTIONS
1. Knowing that the maximum friction force exerted by the bottle on the
cork is 50 N, using virtual work method;
a. Determine the force P that must be applied to the corkscrew to
open the
ME 304 CONTROL SYSTEMS
Spring 2016
Course Instructors
Dr. Tuna Balkan, Dr. Kvan Azgn, Dr. Ali Emre Turgut, Dr. Yiit Yazcolu
MIDTERM EXAMINATION I
April 15, 2016
Time Allowed: 100 minutes
Closed Notes and Books
All questions are equally weighted
All necess
ME 210 Applied Mathematics for Mechanical Engineers
System of Linear Algebraic Equations
Consider a system of m linear algebraic equations with n unknowns: x1, x2, . , xn
where aij and bi are constants.
a11 x1 + a12 x 2 + . + a1n x n = b1
a21 x1 + a22 x 2
ME 210 Applied Mathematics for Mechanical Engineers
Similar Matrices, Similarity Transformation and Diagonalization
Two square matrices [A] and [B] of the same size n are called similar if there exits a
non-singular [P] matrix of size n such that the matr
ME 210 Applied Mathematics for Mechanical Engineers
Inverse of a Square Matrix
Adj [A]
[A]-1 =
det A
Adjoint matrix of A
Determinant of A
Adj [A] = [Cij]T
Transpose of the Cofactor Matrix, [Cij]T
Cofactor Matrix: Matrix formed from cofactors of elements (
ME 210 Applied Mathematics for Mechanical Engineers
Gauss(ian) Elimination
This is a systematic method in which (the coefficient of the) 1st variable is eliminated
(made equal to zero) from the (m1) equations first, the 2nd variable is eliminated
from the
ME 210 Applied Mathematics for Mechanical Engineers
Eigenvalues and Eigenvectors
In a large number of physical and technical problems, the system of linear
equations of the type
a11 x1 + a12 x2 + + a1n xn = x1
a21 x1 + a22 x2 + + a2n xn = x2
.
an1 x1 + an
ME 210 Applied Mathematics for Mechanical Engineers
Cramers Rule
A useful application of determinants to the solution of linear algebraic equations
[A] [x] = [b]
is the Cramers rule, which can be used only when the coefficient matrix [A] is a
non-singular
ME 210 Applied Mathematics for Mechanical Engineers
10. If [C] = [A][B] where [A], [B], and [C] are all square matrices of the same
size, then det[C] = det[A] det[B]
Submatrices and Rank
Submatrix: Any matrix obtained by deleting some rows and/or columns
ME 307 MACHINE ELEMENTS I FALL 2013
HOMEWORK III - SOLUTION
DESIGN OF DETACHABLE JOINTS
Prepared by Hseyin Enes Salman
E-mail: [email protected]
Room: G157 Phone: 5245
In the figure below, a beam is attached to a base by using two M14 x 1.5 ISO 10.9 clas
Middle East Technical University
Department of Mechanical Engineering
ME 312
SPRING 2016
HOMEWORK #1
Assigned: 11.03.2016
Due: 18.03.2016 (until 17:00 to T.A. Eyll imek, A-145)
Guidelines for homework submission:
1. You are encouraged to work in groups of
Middle East Technical University
Department of Mechanical Engineering
ME 312
SPRING 2016
HOMEWORK #2
Assigned: 25.03.2016
Due: 01.04.2016 (until 17:00 to T.A. Serhat Bilyaz, A-146)
Guidelines for home work submission:
1. You are encouraged to work in grou
CS 412
MATLAB Reference Sheet
UW-Madison
Useful workspace functions
help
lookfor
more
load
save
obtain help generally or for a specific function
obtain one-line help if it exists
toggles pagination, useful for longs helps
read variables from a file
save a