Assignment 2
Submit Thursday, February 18, 2010
Circle your final answers
1. By using SVD find all solutions of
1 2 3 x1 19
4 5 6 x = 49
2
7 8 9 x3 79
Solve the problem by a second method.
2. Find all solutions of
5
11
17
23
23 x1 254
25 39 53 x2
Assignment 1
Submit Tuesday, February 9, 2010
1. By using Newtons method find a solution to
x3 e x + 1 = 0
2. By using Newtons method find a solution to
x4 + e x + 1 = 0
3. By using Newtons method find a solution to
x3 e y + 1 = 0
2
xy + y + 2 = 0
4. By
DeterminatiOn of structural modes via the Prony model: System
order and noise induced poles
S. Brauna) and Y. M. Ram
Technion-Israel
Institute Technology,
of
Faculty Mechanical
of
Engineering,
Haifa,Israel
(Received November
2
1985;accepted publication No
Finite Elements in Two Dimensions
The bar truss structure shown in the figure below consists of 3 bars A,B,C, 3 pins 1,2,3,
and two supports. An external force F with components FX and FY is applied to node
2. The bars are uniform with modulus of elastici
The Euler-Lagrange Equation in Expanded Form
F d F
=0
y dx y
(1)
F 2 F dx 2 F dy 2 F dy
=0
y xy dx yy dx y y dx
(2)
F 2 F
2 F dy 2 F d 2 y
=0
y xy yy dx y 2 dx 2
(3)
Special Case I: F = F ( y, y ')
If
F = F ( y , y ') .
(4)
Then the Euler-Lagrange Equ
Calculus of Variation
1. The Mathematical Tool - Taylor Expansion
One-dimensional
f ( x + ) = f ( x)+
2
3
( f x ) + f ( x ) + .
(
f x ) +
1!
2!
3!
Multiple-dimensional
f ( x1 + 1 , x2 + 2 ,., xn + n ) = f ( x1 , x2 ,., xn ) +
+
i f ( x1 , x 2 ,., xn )
xi
10. Numerical integration
10.1. Gauss Quadrature (Due to Golub, Welsch, Math. Comp. 23, pp. 221-230)
Suppose
0
1
2
4(1) 1
A=
1
4(1) 1
2
2
0
4( 2 ) 1
2
2
0
4( 2 ) 1
2
O
3
2
4( 3) 1
O
O
n 1
0
2
4( n 1) 1
and let
A = UU T
be the spectral decomposition of A
7. Polynomial roots finding
7.1. Observer form
Consider the monic polynomial
Pn ( x ) = c0 + c1 x + c2 x 2 + L + cn 1 x n 1 + x n .
Its roots are the eigenvalues of the observer matrix
c0
0
1 0
c1
10
c2
Z=
.
10
c3
OO
M
1 cn 1
Proof. The eigenva
5. The LU factorization
5.1.1 The factorization
If the n 1 leading principal minors of A nn are all non-singular then there
exist L and U such that
A = LU ,
where L is a lower triangular matrix with unit diagonal elements
1
l
1
21
L=
M M O
l n1 l n2 L
0
1
1. Singular Value Decomposition (SVD)
Unless stated to the contrary, m and n are integers, and moreover m n .
1.1. The matrix A mn , can be decomposed into
A = UV T
where
U T U = UU T = I m , = diag cfw_ i 0, i = 1,2,., n mn , and V T V = VV T = I n .
Mor
Assignment 9
Submit Thursday May 6, 2010
1. A force F is applied to the five bar structure as shown in the figure. Determine by
using the finite element method the deflections of the nodes. They bars are cylindrical
with diameter d = 0.05 m and they are m
Assignment 8
Submit Tuesday May 4, 2010
1. Approximate the solution of
y + y = x 3 , 0 < x < 1 ,
y (0 ) = 0 = y (1) .
by using 10 linear finite elements. Compare your solution to the exact result.
Assignment 7
Submit Thursday April 29, 2010
1. Approximate the first 5 eigenvalues of
y + y = 0
y (0 ) = y( ) = 0
by using Rayleigh-Ritz method with 5 admissible functions. Compare the
approximation to the exact solution
2. Solve
e x y + e x + 2 y = 1
Assignment 6
Submit Tuesday April 27, 2010
A uniform cable of length L = 5 is pinned at (x A = 0 y A = 0 ) and (x A = 2
Determine and plot the shape of the cable at static equilibrium position.
y A = 2 ) .
Assignment 5
Submit April 15, 2010
1. Write a program that evaluate the Gauss Quadratures and weights for approximation
of order n. By this program evaluate
1
xe dx
x
1
with n=3, 5, 10. Compare your solution to the exact result. Present your solution in
Assignment 4
Submit Tuesday, March 25, 2010
1. By using model over-determination estimate the model order n and the coefficients
ck , sk of the following function,
n
f = g + ck e s k x
k =1
where g represents noise. The data for this assignment are given
Assignment 3
Submit Tuesday, March 16, 2010
Circle your final answer
1. By using LU factorization (with pivoting) find all solutions of
0 1 2 3 x1 5
4 5 6 7 x 13
2 =
8 9 10 11 x3 21
12 13 14 15 x4 29
(Solve by hand)
2. By using bisection find t