Introduction
Linear Equations
Gauss Elimination
Elimination
Jungyul Park
Park
Department of Mechanical Engineering
of Mechanical Engineering
Sogang University
Linear Equations?
A linear equation in n variables x1,x2,xn:
a1x1 + a2x2 + + anxn = b
x1, x2, ,
Problem 5.17
The data listed below are claimed for a power cycle operating
between hot and cold reservoirs at 1000K and 300K. For each
case, determine whether the cycle operates reversibly, operate
s irreversibly, or is impossible
a)QH = 600 kJ, Wcycl= 30
<7 >
- : 5 18
1. The data listed below are claimed for a power cycle operating between hot and cold reservoirs
at 1000 K and 300 K, respectively. For each case, determine whether the cycle operates reversibly,
operates irreversibly, or is impossible.
(a)
EXAMPLE
Steam enters a turbine operating at steady-state with a mass flow rate
of 4600 kg/h. The turbine develops a power output of 1000 kW. At
the inlet the pressure is 60 bars, the temperature is 400C and the
velocity is 10 m/s. At the exit the pressure
Problem 4-23
Steam enters a horizontal pipe operating at st
eady state with a specific enthalpy of 3000 kJ/
kg and a mass flow rate of 0.5 kg/s. At the exi
t, the specific enthalpy is 1700 kJ/kg. If there i
s no significant change in kinetic energy from
i
Problems and Solutions Section 8.2 (8.8 through 8.20)
8.8
Consider the bar of Figure P8.3 and model the bar with two elements. Calculate
the frequencies and compare them with the solution obtained in Problem 8.3.
Assume material properties of aluminum, a
Chapter 8
Problems and Solutions Section 8.1 (8.1 through 8.7)
8.1
Consider the one-element model of a bar discussed in Section 8.1. Calculate the
finite element of the bar for the case that it is free at both ends rather than
clamped.
Solution: The finit
Problems and Solutions Section 8.3 (8.21 through 8.33)
8.21
Use equations (8.47) and (8.46) to derive equation (8.48) and hence make sure
that the author and reviewer have not cheated you.
Solution:
u( x, t ) = C1 (t ) x 3 + C2 (t ) x 2 + C3 (t ) x + C4 (
Problems and Solutions Section 8.4 (8.34 through 8.43)
8.34
Refer to the tapered bar of Figure P8.13. Calculate a lumped-mass matrix for this
system and compare it to the solution of Problem 8.13. Since the beam is tapered,
be careful how you divide up th
Problems and Solutions Section 8.5 (8.44 through 8.49)
8.44
Derive a consistent-mass matrix for the system of Figure 8.9. Compare the
natural frequencies of this system with those calculated with the lumped-mass
matrix computed in Section 8.5.
Solution: U
Problems and Solutions Section 8.6 (8.50 through 8.53)
8.50
Consider the machine punch of Figure P8.15. Recalculate the fundamental
natural frequency by reducing the model obtained in Problem 8.16 to a single
degree of freedom using Guyan reduction.
Solut
Problems and Solutions Section 7.2 (7.1-7.5) 7.1 A low-frequency signal is to be measured by using an accelerometer. The signal is physically a displacement of the form 5 sin (0.2t). The noise floor of the accelerometer (i.e. the smallest magnitude signal
Problems and Solutions Section 7.3 (7.6-7.9) 7.6 Represent 5 sin 3t as a digital signal by sampling the signal at /3, /6 and /12 seconds. Compare these three digital representations. Solution: Four plots are shown. The one at the top far right is the exac
Problems and Solutions for Section 7.4 (7.10-7.19) 7.10 Consider the magnitude plot of Figure P7.10. How many natural frequencies does this system have, and what are their approximate values?
100
Magnitude
10 10 10 10
1
2
3
4
0
10
20 (Hz)
30
40
50
1
Magni
Problems and Solutions Section 7.5 (7.20-7.24) 7.20 Using the definition of the mobility transfer function of Window 7.4, calculate the Re and Im parts of the frequency response function and hence verify equations (7.15) and (7.16). Solution: From Window
5.1.3 and , trace , determinant .
. The vector is in the nullspace so .
5.1.5
5.1.9 Transpose :
5.1.12 The eigenvalues of are .
5.1.17 and and all have the same eigenvectors. The eigenvalues are
and for , 1 and for , 1 and 0 for . Therefore is h
4.2.2 and and and .
4.2.5 (singular); ; ; ; (2
exchanges).
4.2.14 Adding every column of to the first column makes it a zero column, so
. If every row of adds to , then every row of adds to and
. But need not be :
has , but
.
4.2.15 (a) Rule 3 (facto
Gauss Elimination
Jungyul Park
Department of Mechanical Engineering
Sogang University
Review and Summary
To solve
Geometry of linear equations
Row pictures : Intersection of planes
Column pictures : Combination of columns
Limitation to the geometric appro
Matrix notation and
matrix multiplication
Jungyul Park
Department of Mechanical Engineering
Sogang University
Review and Summary
Gauss elimination
Simplify our set of linear equations Row
echelon form
Gauss-Jordan elimination
Simplify our set of linear eq
Vector Spaces
Jungyul Park
Department of Mechanical Engineering
Sogang University
LU-Factorization
Note that L is square matrix, it has the same
number of rows as A and U.
A=LU
Lower triangular
Product of
inverses elementary
matrices
Upper triangular
Vector Spaces II
Jungyul Park
Department of Mechanical Engineering
Sogang University
Definition of Linear
Independent
The vectors v1,v2,vn in a vector space V
are said to be linearly independent if
1v1+2v2+nvn=0
has exactly one solution 1=2=n=0.
Example:
Vector Spaces III
Vector Spaces III
Linear Transformations
Jungyul Park
Park
Department of Mechanical
Engineering
Engineering
Sogang University
Linear Transformations
(Definition)
Definition: V and W are vector spaces. A
function
function L:VW is called a
Determinants
Jungyul Park
Department of Mechanical Engineering
Sogang University
Notes for determinants
The determinant is a scalar-valued function defined on the set of
square matrices.
The main use of determinants is to compute and establish properties
Eigenvalues and
Eigenvectors I
Jungyul Park
Department of Mechanical Engineering
Sogang University
Eigenvalues and Eigenvectors
Definition: Let A be an nn matrix. A scalar is
called an eigenvalues of A if there exist a
nonzero vector x such that Ax=x. The
Eigenvalues and
Eigenvectors II
Jungyul Park
Department of Mechanical Engineering
Sogang University
Diagonalization (review)
Definition: An nn matrix A is said to be diagonalizable if
there exist a nonsigular matrix X and a diagonal matrix D
such that X-1
Positive Definite
Jungyul Park
Department of Mechanical Engineering
Sogang University
Example of Jordan Form
Reduce A to Jordan Form:
1 2 3 1 . Check the nullity of (A - I),
Find X1
Example of Jordan Form
Find x2 and x3 (the generalized eigenvectors),
Use
4.1
SOLUTIONS
Notes: This section is designed to avoid the standard exercises in which a student is asked to check ten axioms on an array of sets. Theorem 1 provides the main homework tool in this section for showing that a set is a subspace. Students sho
5.1
SOLUTIONS
Notes: Exercises 16 reinforce the definitions of eigenvalues and eigenvectors. The subsection on
eigenvectors and difference equations, along with Exercises 33 and 34, refers to the chapter introductory example and anticipates discussions of
1.2.2 No solution; zero tight sides produce 3 lines through origin; right sides
give and ; will give and ; any combination is solvable.
1.2.6 Both and give a line of solutions. All other give , .
1.2.8 The conditions for a straight line can be written as
2.1.2 (a), (d), (e) are subspaces.
2.1.4 (b), (d), (e) are subspaces. Cant multiply by -1 in (a) and (c). Cant add in (f).
2.1.8 The solutions form a line, subspace, nullspace of
2.1.13
3
4
20 . Rule 8 is broken: If
3
is different from
then
00
;
00
consis