Engineering Mathematics I -...
Transcript of Engineering Mathematics I -...
Bong-Kee Lee School of Mechanical Systems Engineering
Chonnam National University
Engineering Mathematics I
4. Systems of ODEs. Phase Plane. Qualitative Methods
School of Mechanical Systems Engineering Engineering Mathematics I
4.0 Basics of Matrices and Vectors
행렬(matrix)
– 요소(entry), 행(row), 열(column), 정방행렬(square matrix)
– 열벡터(column vector), 행벡터(row vector)
nnnn
n
n
jk
aaa
aaa
aaa
aA
21
22221
11211
nx
x
x
x
2
1
nvvvv
21
School of Mechanical Systems Engineering Engineering Mathematics I
4.0 Basics of Matrices and Vectors
행렬과 벡터의 연산
– 상등성(equality)
– 덧셈(addition), 스칼라곱(scalar multiplication), 행렬의 곱(matrix multiplication)
2222212112121111
2221
1211
2221
1211,,,, babababaBA
bb
bbB
aa
aaA
22222121
12121111
2221
1211
2221
1211
baba
baba
bb
bb
aa
aaBA
2221
1211
2221
1211
acac
acac
aa
aaccA
n
m
mkjmjk bacABC1
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aaAB
School of Mechanical Systems Engineering Engineering Mathematics I
4.0 Basics of Matrices and Vectors
행렬과 벡터의 연산
– 전치(transposition)
– 역행렬(inverse of a matrix), 단위행렬(unit matrix)
21
2
1
2212
2111
2221
1211, vvv
v
vv
aa
aaA
aa
aaA TT
AIAAI
I
10
01
IAAAAABIBAAB 111
1112
2122
211222111112
21221
2221
1211
1
det
1
aa
aa
aaaaaa
aa
AA
aa
aaA
School of Mechanical Systems Engineering Engineering Mathematics I
4.0 Basics of Matrices and Vectors
행렬과 벡터의 연산
– 1차 독립(linearly independent)
– 고유값(eigenvalue), 고유벡터(eigenvector)
0
0
21
2
2
1
1
n
n
n
ccc
vcvcvc
xxA
eigenvalue eigenvector
0det
0
0
IA
xIA
xxA
characteristic equation
eigenvalue
eigenvector
School of Mechanical Systems Engineering Engineering Mathematics I
4.0 Basics of Matrices and Vectors
행렬과 벡터의 연산
– 고유값(eigenvalue), 고유벡터(eigenvector)
8.0
1
64.1
1or
8.0
1
8.0ii
1
2
5
1or
1
2
1
22
20
0
2.36.1
42
2.36.1
0.40.2
2i
8.0,2
06.18.22.16.1
0.40.4det
2.16.1
0.40.4
example
2
2
2
2
2
1
21
21
21
2
1
1
21
2
xx
xxxx
x
x
xx
xxxx
xx
x
xxIA
IAA
characteristic equation eigenvalues
eigenvector
eigenvector
School of Mechanical Systems Engineering Engineering Mathematics I
4.0 Basics of Matrices and Vectors
연립미분방정식(systems of DEs)
– 두 개 이상의 미지함수를 갖는 두 개 이상의 상미분방정식
– 미분 • 요소(또는 성분)가 변수인 행렬(또는 벡터)의 도함수는 각각의 요
소를 미분하여 구함
2221212
2121111
'
'
yayay
yayay
nnnnnn
nn
nn
yayayay
yayayay
yayayay
2211
22221212
12121111
'
'
'
yAy
y
aa
aa
yaya
yaya
y
yy
yayay
yayay
2
1
2221
1211
222121
212111
2
1
2221212
2121111
'
''
'
'
t
e
ty
tyty
t
e
ty
tyty
tt
cos
2
'
''
sin
2
2
12
2
1
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
2개 탱크에서의 혼합 문제 탱크 T1과 T2에는 초기에 각각 물 100갤런이 들어 있다. 탱크 T1에는 순수한 물만 들어 있는 반면 탱크 T2에는 150파운드의 비료가 용해되어 있다. 양쪽 탱크의 액체를 분당 2갤런(2 gal/min)의 속도로 순환시키며 고루 섞어주면 탱크 T1의 비료의 양 y1(t)와 탱크 T2
의 비료의 양 y2(t)는 시간 t에 따라 변하게 될 것이다. 탱크 T1의 비료의 양이 적어도 탱크 T2에 남아 있는 비료의 양의 반이 되기 위해서는 얼마 동안 액체를 순환시켜야 하는가?
212
121
gal100
1gal/min2
gal100
1gal/min2outflowinflow'
gal100
1gal/min2
gal100
1gal/min2outflowinflow'
yyy
yyy
yAy
Ay
yy
yyy
yyy
'
02.002.0
02.002.0,
02.002.0'
02.002.0'
2
1
212
211
vector equation
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
2개 탱크에서의 혼합 문제
xxA
exAexAyAy
ex
x
ex
ex
y
yexy
ex
exe
x
x
y
yexy
tt
t
t
t
t
t
t
tt
'&
'
''
assumption
2
1
2
1
2
1
2
1
2
1
2
1
eigenvalue problem
04.0,0
004.002.002.0
02.002.0det
IA
eigenvalues
1
1
1
1
0
0
02.002.0
02.002.0
04.0ii
1
1
1
1
0
0
02.002.0
02.002.0
0i
2
1
1
1
2
1
21
2
1
2
1
1
1
1
2
1
21
2
1
1
x
xx
x
x
xxxx
x
xxIA
x
xx
x
x
xxxx
x
xxIA
eigenvector
eigenvector
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
2개 탱크에서의 혼합 문제
min5.27
3ln04.03
1507575
505.0out find
7575
7575or
1
175
1
175
75,75150
0
1
1
1
10
1500,00conditions initial
1
1
1
1
principleion superposit
04.004.0
1
121
04.0
04.0
2
104.0
21
21
21
21
21
04.0
21
2
2
1
121
t
teey
yyy
e
e
y
yey
cccc
ccccy
yy
eccexcexcy
exy
tt
t
t
t
ttt
t
general solution
about 30min.
particular solution
ty1
ty2
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
전기회로망 아래와 같은 전기회로망에서 전류 I1(t)와 I2(t)를 구하라. 스위치가 닫히는 순간인 t=0에서 모든 전류와 전하는 0이라 가정한다.
12)(' 2111 IIRLI
01
212122 dtIC
IIRIR
gJAJ
gAI
IJ
III
III
III
III
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8.4
0.12,
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8.42.16.1'
1244'
4.0'4.0'
1244'
1
1
212
211
212
211
nonhomogeneous
ph JJJ
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
전기회로망
tttt
h
t
h
ececexcexcJ
xx
IAexJJ
8.0
2
2
1
2
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1
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2
2
1
1
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1
1
2
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1
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02.16.1
0.40.4
0det:
21
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p
pp
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a
gaAgJAJJ
a
aaJgJ
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0.40.4
0'0'
ectorconstant vectorconstant v8.4
0.12:
2
1
2
1
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
전기회로망
tt
tt
tttt
tt
tt
tttt
ph
eeI
eeI
eeaexexJ
ccII
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1
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2
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12
8.0
2
2
11
8.0
2
2
1
2
2
1
1
44
358or
0
3
4
5
4
854
5,4000conditions initial
8.0
32or
0
3
8.0
1
1
221
general solution
particular solution
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
전기회로망
– 상평면 표현(phase plane representation)
tI1
tI2
1I
2I
상평면(phase plane): I1-I2 plane
궤적(trajectory)
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
상미분방정식의 변환
– n계 상미분방정식 → 1계 연립 상미분방정식
1,,',, nn yyytFy
1
21 ,,', n
n yyyyyy
nn
nn
yyytFy
yy
yy
yy
,,,,'
'
'
'
21
1
32
21
n-th order ODE
n 1st order ODEs
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
상미분방정식의 변환
– 예: 용수철에 매달린 물체
01
det
10
'
''
'
'
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'''or 0'''
2
2
1
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2
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1
212
21
2
21
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1
m
k
m
c
m
k
m
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m
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yAy
y
m
c
m
ky
m
cy
m
k
y
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yy
y
yy
ym
cy
m
ky
yy
ym
ky
m
cyy
yyy
yy
yy
ym
ky
m
cykycymy
characteristic equation
School of Mechanical Systems Engineering Engineering Mathematics I
4.1 Systems of ODEs as Models
상미분방정식의 변환
– 예: 용수철에 매달린 물체
tt
tt
tt
tt
tttt
ececy
ecec
ecececec
y
y
y
yy
ececexcexcy
xxxx
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xxxx
xxIA
m
k
m
c
kcm
5.1
2
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1
5.1
2
5.0
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5.1
2
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15.1
2
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1
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2
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1
2
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1
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1
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2
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1
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11
21
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1
202
0
0
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15.0:5.0i
5.1,5.005.15.075.020
75.0,2,1 of case for the
21
School of Mechanical Systems Engineering Engineering Mathematics I
4.2 Basic Theory of Systems of ODEs
연립 상미분방정식
– 해 벡터(solution vector) • 어떤 구간 a<t<b 에서 연립 상미분방정식을 만족하는 미분가능한
n개의 함수들의 집합
– 초기조건(initial condition)
nnn
n
n
yyytfy
yyytfy
yyytfy
,,,,'
,,,,'
,,,,'
21
2122
2111
',,','
&
,,,
21
21
n
n
yyy
yyy
nn f
f
f
y
y
y
11
,
ytfy
,'
Tnnn ththhthythy
111 ,,
Tnnn KKKKtyKty
10101 ,,
n개의 미지함수
n개의 미지함수의 도함수
School of Mechanical Systems Engineering Engineering Mathematics I
4.2 Basic Theory of Systems of ODEs
연립 상미분방정식
– 존재성과 유일성 정리
unique. issolution thisand ,conditions initialgiven thesatisfying
interval someon solution a has ODEs of system Then the .
point thecontaining space of domain somein
sderivative partial continuous having functions continuous be Let
theoremuniqueness and existence
0010
211
1
1
1
ttt,K,,Kt
yytyRy
f,,
y
f,,
y
f
,f,f
n
n
n
n
n
n
School of Mechanical Systems Engineering Engineering Mathematics I
4.2 Basic Theory of Systems of ODEs
선형연립방정식(linear system)
– 연립 상미분방정식이 y1,…,yn에 대하여 1차일 경우
nnn
n
n
yyytfy
yyytfy
yyytfy
,,,,'
,,,,'
,,,,'
21
2122
2111
tgytaytay
tgytaytay
nnnnnn
nn
11
111111
'
'
gyAy
'
nnnnn
n
g
g
g
y
y
y
aa
aa
A
11
1
111
,,
homogeneous
nonhomogeneous
0g
0g
School of Mechanical Systems Engineering Engineering Mathematics I
4.2 Basic Theory of Systems of ODEs
선형연립방정식(linear system)
– 선형인 경우의 존재성과 유일성
– 중첩의 원리 or 선형성 원리
unique. issolution this
and ,conditions initial thesatisfying interval on this solution a has
ODEs of systemlinear Then the .point thecontaining
intervalopen an on of functions continuous be s' and s' Let the
caselinear in the uniqueness and existence
0
ty
ttt
tga jjk
.n combinatiolinear any is so interval,
someon systemlinear shomogeneou theof solutions are and If
principlelinearity or principleion superposit
2
2
1
1
21
ycycy
yy
School of Mechanical Systems Engineering Engineering Mathematics I
4.2 Basic Theory of Systems of ODEs
선형연립방정식(linear system)
– 기저(basis) or 기본시스템(fundamental system) • 어떤 구간에서 1차 독립인 n개의 해의 집합
– 일반해(general solution): 기저들의 일차결합
– 기본행렬(fundamental matrix)과 Wronskian • n개의 해를 열로 가지는 행렬
nn yyyy
,,,, 121
n
n
n
n ycycycycy
1
1
2
2
1
1
n
nnn
n
n
n
yyy
yyy
yyy
yyY
21
2
2
2
1
2
1
2
1
1
1
1
tindependenlinearly :,,0
det
1 nyyW
YW
cYy
cccT
n
1
Wronskian
general solution fundamental matrix
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상수계수를 갖는 연립방정식
– 제차 선형연립방정식 • 상수계수 → A의 모든 요소가 t에 의존하지 않음
yAy
'
eigenvalue problem
texy
xxA
exAyA
exexy
t
tt
''
tnnt nexyexy
,,111
tn
n
t nexcexcy
11
1
assumption
general solution
0det IA
characteristic equation
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 매개변수표현(매개변수방정식) • t를 매개변수로 이용하여 하나의 곡선으로 표현
• 궤적(trajectory) (궤도(orbit), 경로(path))
• 상평면(phase plane): y1-y2 평면
• 상투영(phase portrait): 상평면을 궤적으로 채움
222121
212111
2
1
2221
1211
2
1
'
''
yaya
yaya
y
y
aa
aa
y
yyAy
ty
tyy
2
1
t
tyty 21 &
ty1
ty2
or
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 상투영: 상평면에서의 궤적
tt
tt
ecec
excexcy
xx
IA
yyy
yyyyyAy
4
2
2
1
2
2
1
1
21
21
212
211
1
1
1
1
1
1,
1
1
4,2
031
13det
3'
3'
31
13'
example
21
c1=0
c2=0
→ 해 전체를 정성적으로 파악하는데 유용함
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 임계점(critical point): dy2/dy1 값이 정의되지 않는 점
– 임계점의 종류 • 비고유마디점(improper node)
• 고유마디점(proper node)
• 안장점(saddle point)
• 중심(centers)
• 나선점(spiral point)
222121
212111
2
1
2221
1211
2
1
'
''
yaya
yaya
y
y
aa
aa
y
yyAy
222121
212111
1
2
1
2
1
2
'
'
'
'
yaya
yaya
y
y
dty
dty
dy
dy
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 비고유마디점(improper node) • 임계점에서 두 개의 궤적을 제외한 모든 궤적이 같은 접선의 극한
방향을 갖는 경우
tt ececy
xx
IA
yyAy
4
2
2
1
21
21
1
1
1
1
1
1,
1
14,2
031
13det
31
13'
example
slope: [1 1]T
(as t→∞, y2 goes to 0 faster)
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 고유마디점(proper node) • 임계점에서 모든 궤적들이 각각 한정된 극한방향을 가지는 경우
로 임의로 주어진 방향에 대하여 그 방향을 극한방향으로 가짐
tt ececy
xx
IA
yyAy
1
0
0
1
1
0,
0
11
010
01det
10
01'
example
21
21
21
1221
2211 &
ycyc
ecyecy tt
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 안장점(saddle point) • 임계점에서 두 개의 들어오는 궤적과 두 개의 나가는 궤적이 존재
하고 임계점 근방의 모든 다른 궤적들은 임계점을 우회하는 경우
tt ececy
xx
IA
yyAy
1
0
0
1
1
0,
0
11,1
010
01det
10
01'
example
21
21
21
constccyy
ecyecy tt
2121
2211 &
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 중심(centers) • 무한히 많은 닫힌 궤적에 의해 둘러싸여진 임계점
itit ei
cei
cy
ix
ixii
IA
yyAy
2
2
2
1
21
21
2
1
2
1
2
1,
2
12,2
04
1det
04
10'
example
constyy
yyyy
yyyy
2
2
2
1
2211
1221
2
12
''4
4'&'
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 나선점(spiral point) • t→∞에 따라 궤적들이 임계점 근방에서 나선형을 그리며 임계점
에 접근 또는 임계점에서 멀어지는 경우
titi ei
cei
cy
ix
ixii
IA
yyAy
1
2
1
1
21
21
11
1,
11,1
011
11det
11
11'
example
tcer
tryy
yyyyyy
yyyyyy
,,scoordinatepolar
''
'&'
21
2
2
2
12211
212211
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 고유벡터가 기저를 형성하지 않는 경우 • 퇴화마디점(degenerate node)
xuIAeuAeuex
euAtexeuAtexAyAyeutexexy
texyeutexy
xxx
xxIA
IA
yyAy
ttt
ttttttt
ttt
222
22
21
2
1
21
2
'&'
!assumption
?,1
1
11
113
0321
14det
21
14'
example
real double root
important !
School of Mechanical Systems Engineering Engineering Mathematics I
4.3 Constant-Coefficient Systems. Phase Plane Method
상평면법(phase plane method)
– 고유벡터가 기저를 형성하지 않는 경우 • 퇴화마디점(degenerate node)
tt
tt
tt
ttt
etcec
tecec
etcec
eutexcexcycycy
u
uu
u
u
xuIA
3
2
3
1
3
2
3
1
3
2
3
1
1
2
1
1
2
2
1
1
2
1
2
1
1
1
0
1
1
1
1
1
0
1
1
321
134
general solution
School of Mechanical Systems Engineering Engineering Mathematics I
4.4 Criteria for Critical Points. Stability
임계점에 대한 판별기준
2
21
21
21
21
2
21
21
21
2
2
2
2211
2211
2
211222112211
2
2221
1211
2221
1211
4
1
2
1,
2
1
22
40
4
det
0det
0det
'
q
p
qp
p
pp
pqppqp
qp
Aq
aap
Aaa
aaaaaaaa
aaIA
yaa
aayAy
School of Mechanical Systems Engineering Engineering Mathematics I
4.4 Criteria for Critical Points. Stability
임계점에 대한 판별기준
– 임계점에 대한 고유값 판별기준(Table 4.1)
임계점의 종류 p q Δ λ1 & λ2
(a) 마디점 q > 0 Δ ≥ 0 실수, 같은 부호
(b) 안장점 q < 0 실수, 반대 부호
(c) 중심 p = 0 q > 0 순허수
(d) 나선점 p ≠ 0 Δ < 0 복소수(순허수
가 아님)
School of Mechanical Systems Engineering Engineering Mathematics I
4.4 Criteria for Critical Points. Stability
안정성(stability)
– 임계점의 분류하기 위한 또 다른 방법
– 안정성 • 어느 순간의 물리적 시스템의 작은 변화(작은 외란)가 이후의 모
든 시간에서 시세틈의 거동에 단지 작은 변화만을 주는 것
– 임계점의 안정성 분류 • 안정적 임계점(stable critical point)
• 불안정적 임계점(unstable critical point)
• 안정적 흡인 임계점(stable and attractive critical point)
School of Mechanical Systems Engineering Engineering Mathematics I
4.4 Criteria for Critical Points. Stability
안정성(stability)
– 임계점에 대한 안정성 판별 기준(Table 4.2)
안정성의 형태 p q
(a) 안정적 흡인 임계점 p < 0 q > 0
(b) 안정적 임계점 p ≤ 0 q > 0
(c) 불안정적 임계점 p > 0 q < 0 or
School of Mechanical Systems Engineering Engineering Mathematics I
4.4 Criteria for Critical Points. Stability
안정성(stability)
– 용수철에 달린 물체의 자유운동
m
k
m
c
m
kq
m
cp
m
k
m
c
m
c
m
kIA
y
m
c
m
ky
y
m
c
m
ky
m
cy
m
k
y
y
yy
y
y
y
yy
ym
ky
m
cykycymy
4,,01
det
1010
''
''
'
0'''0'''
2
2
2
1
21
2
2
1
(1) no damping (c=0): p=0, q>0 → center (2) underdamping (c2<4mk): p<0, q>0, Δ<0 → stable & attractive (spiral point) (3) critical damping (c2=4mk): p<0, q>0, Δ=0 → stable & attractive (node) (4) overdamping (c2>4mk): p<0, q>0, Δ>0 → stable & attractive (node)
School of Mechanical Systems Engineering Engineering Mathematics I
4.5 Qualitative Methods for Nonlinear Systems
비선형연립방정식에 대한 정성법
– 정성법(qualitative method) • 방정식의 해를 실제로 구하지 않으면서 해에 대한 정성적인 정보
를 얻는 방법
• 연립방정식의 해를 해석적으로 구하기 어렵거나 불가능한 경우에 유용함
• 비선형연립방정식(nonlinear system)이 가지는 해의 성질을 이해하는데 많은 도움을 줌
2122
2111
,'
,''
yyfy
yyfyyfy
School of Mechanical Systems Engineering Engineering Mathematics I
4.5 Qualitative Methods for Nonlinear Systems
비선형연립방정식에 대한 정성법
– 비선형연립방정식의 선형화(linearization) • 임계점 근처에서의 비선형연립방정식을 선형연립방정식으로 변
환
2221212
2121111
2122221212
2112121111
2122
2111
'
'
'
,'
,'
'
,'
,'
'
yayay
yayay
yAy
yyhyayay
yyhyayay
yhyAy
yyfy
yyfy
yfy
point. spiral aor (3) aspoint critical of kind
same thehavemay (1) then s;eigenvalueimaginary pureor equal has ifoccur Exceptions
(3). of system linearized theof thoseas same theare (1) ofpoint
critical theofstability and kind then the(2),in 0det if and ,0,0:point critical theof
odneighborho ain sderivative partial continuous have and continuous are (1)in and If
ionlinearizat
0
21
A
AP
ff
(1) (2) (3)
School of Mechanical Systems Engineering Engineering Mathematics I
4.5 Qualitative Methods for Nonlinear Systems
자유비감쇠진자의 선형화
L
gkk
LmmgamF
where0sin''
''sin
tkBtkAk sincos0''
sin0
,2,1,0 where0,:0''sin'''
''
'&
21
2
21
21
nnyyky
yy
yy
critical points
mathematical model
nonlinear
04,0/,00
10'
6
1sinsin0,0for
,0,4,0,2,0,01
1
3
111
kLgkqpyk
yAy
yyyy
→ center
School of Mechanical Systems Engineering Engineering Mathematics I
4.5 Qualitative Methods for Nonlinear Systems
자유비감쇠진자의 선형화
→ saddle point
04,0/,00
10'
6
1sinsinsin
'',0,for
,0,3,0,2
1
3
1111
21
kLgkqpyk
yAy
yyyyy
yy
School of Mechanical Systems Engineering Engineering Mathematics I
4.5 Qualitative Methods for Nonlinear Systems
감쇠진자의 선형화
L
gkkc
LmcmgamF
where0sin'''
'''sin
yck
yAy
yck
yAy
10'
,0,3,0,2
10'
,0,4,0,2,0,01
→ spiral point (c>0)
→ saddle point (c>0)
School of Mechanical Systems Engineering Engineering Mathematics I
4.5 Qualitative Methods for Nonlinear Systems
Lotka-Volterra 개체군 모델
– 포식자-먹이 개체군 모델: 눈토끼(y1) vs. 스라소니(y2)
2212122
2112111
,'
,'
lyykyyyfy
ybyayyyfy
School of Mechanical Systems Engineering Engineering Mathematics I
4.6 Nonhomogeneous Linear Systems of ODEs
비제차 선형연립방정식
0
0
'
g
g
gyAy
homogeneous
nonhomogeneous
gyAyy
yAyy
yyy
p
h
ph
'
'
general solution
(i) method of undetermined coefficients (ii) method of variation of parameters
School of Mechanical Systems Engineering Engineering Mathematics I
4.6 Nonhomogeneous Linear Systems of ODEs
비제차 선형연립방정식
– 미정계수법(method of undertermined coefficients)
2
6
3
322&
1
12
'&
22'
~:ii
1
1
1
1:i
2
6
31
13'example
21
21
2
1222
22
222
222
4
2
2
1
2
vv
vv
v
v
a
agevAeveu
aa
auuuA
gevAteuAgyAy
evteueuy
evteuyegy
ececyy
yyyeygyAy
ttt
ttpp
tttp
ttptp
tthh
pht
School of Mechanical Systems Engineering Engineering Mathematics I
4.6 Nonhomogeneous Linear Systems of ODEs
비제차 선형연립방정식
– 미정계수법(method of undertermined coefficients)
ttttph
ttttph
eteececyyy
vk
eteececyyy
vk
k
kvvva
vv
vv
v
v
a
a
224
1
2
1
224
1
2
1
12
21
21
2
1
2
2
1
12
1
1
1
1
2
2 ,2for
4
0
1
12
1
1
1
1
4
0 ,0for
44,2
2
6
3
32
School of Mechanical Systems Engineering Engineering Mathematics I
4.6 Nonhomogeneous Linear Systems of ODEs
비제차 선형연립방정식
– 매개변수변환법(method of variation of parameters)
tt
tt
tt
tt
ttt
tt
ppp
pp
tt
tt
tt
tt
tthh
pht
ee
ee
ee
ee
eee
eeY
gYuguYguYguAYuYuY
AYyyYyyY
guAYgyAyuYuYy
tutYyy
ctYc
c
ee
ee
ecec
ececececycycyy
yyyeygyAy
44
22
22
44
6
1
42
42
1
1
2121
2
1
42
42
4
2
2
1
4
2
2
14
2
2
1
2
2
1
1
2
2
1
2
1
'''''
'''
' & '''
:ii
1
1
1
1:i
2
6
31
13'example
xyxvxyxuxy
xycxycxy
p
h
21
2211
School of Mechanical Systems Engineering Engineering Mathematics I
4.6 Nonhomogeneous Linear Systems of ODEs
비제차 선형연립방정식
– 매개변수변환법(method of variation of parameters)
tttt
ttttph
tt
ttt
ttt
ttt
tt
p
t
t
t
tt
t
tt
tt
eteecec
eet
tececyyy
eet
t
eete
eete
e
t
ee
eeuYy
e
ttd
eu
ee
e
ee
eegYu
224
2
2
1
424
2
2
1
42
422
422
242
42
20~
2
22
2
44
22
1
2
2
1
12
1
1
1
1
2
2
22
22
1
1
1
1
2
2
22
22
222
222
22
2
22
2~
4
2
4
2
2
6
2
1'