+@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b...

21
τ · ds(t) dt + s(t)= Ke(t) τ K e(t) L −→ E(p) s(t) L −→ S(p) τ · p · S(p)+ S(p)= K · E(p) H(p)= S(p) E(p) = K 1 + τ · p K 1 + τ · p s(t) e(t)= E 0 · H(t) s(t) τ · ds(t) dt + s(t)= Ke(t) s(t)= K · E 0 · 1 e t τ

Transcript of +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b...

Page 1: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

τ · ds(t)dt

+ s(t) = K e(t)τ

K

e(t)L−→ E(p) s(t)

L−→ S(p)

τ · p · S(p) + S(p) = K · E(p)

H(p) =S(p)

E(p)=

K

1+ τ · pK

1+ τ · p

s(t)e(t) = E0 ·H(t)

s(t)

τ · ds(t)dt

+ s(t) = K e(t)

s(t) = K · E0 ·⎛⎝1− e

−t

τ

⎞⎠

Page 2: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

S(p) = H(p) · E(p)S(p) =

K

1+ τ · p ·E0

pe(t)

L−→ E0

p

S(p) =A

p+

B

1+ τ · pS(p) =

K · E0

p−

K · E0 · τ1+ τ · p

K · E0

p

L−1−−→ K · E0 ·H(t)

K · E0 · τ1+ τ · p

L−1−−→ K · E0 · e−t

τ ·H(t)

s(t) = K · E0 ·⎛⎝1− e

−t

τ

⎞⎠H(t)

t→∞ s(t) =p→0

p · S(p)

t→∞ s(t) =p→0

(p · K

1+ τ · p ·E0

p

)

t→∞ s(t) = K · E0

K · E0 t→∞s(t)

t = 0

t→0s(t) =

p→∞p · S(p)

t→0s(t) =

p→∞(p · K

1+ τ · p ·E0

p

)

t→0s(t) = 0

s(t)t = 0

f(t) F(p)

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L

(f(t)

)= p · F(p) − f(0+)

L

(f(t)

)= p · F(p)

s(t) =s(t)

s∞ = K · E00 95 ·K · E0

0 63 ·K · E0

τ3τ

T5 = 3ττ τ1

3τ1τ2

3τ2τ3

3τ3

τ

t→0s(t) =

t→0

s(t)=

p→∞p2 · S(p) =p→∞

(p2 · K

1+ τ · p ·E0

p

)=

K

τ· E0

K · E0

τ=

s∞τ

K · E0 τ

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T5±5

s(∞) − s(T5 )

s(∞)= 0 05

K · E0 −K · E0 ·(1− e

−T5τ

)K · E0

= 0 05

e−T5

τ = 0 05

T5 ≈ 3 · τ

τ

K

τ

τ

K

τ2

e(t) = δ(t)

S(p) = H1(p) =K

1+ τ · p ⇒ s(t) =K

τ· e−

t

τ

τK · a

a

e(t) = a · t ·H(t)

S(p) = H1(p) · E(p) = K

1+ τ · p ·a

p2

s(t) = K · a ·⎛⎝t− τ ·

⎛⎝1− e

−t

τ

⎞⎠⎞⎠

K · a(τ 0)

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a2 ·2 s(t)

2+ a1 · s(t)

+ a0 · s(t) = b0 · e(t)

1

ω2n

·2 s(t)

2+

2 · ξωn

· s(t)+ s(t) = K · e(t)

ωn > 0 rad s−1

ξ > 0 :

K

1

ω2n

·2 s(t)

2+

1

Q ·ωn· s(t)

+ s(t) = K · e(t)ωn > 0

Q =1

2 · ξK

e(t) −→ E(p) s(t)L−→ S(p)

p2

ω2n

· S(p) + 2 · ξωn

· p · S(p) + S(p) = K · E(p)(p2

ω2n

+2 · ξωn

· p+ 1

)· S(p) = K · E(p)

H(p) =S(p)

E(p)=

K

1+2 · ξωn

· p+p2

ω2n

K

1+2 · ξωn

· p+p2

ω2n

Δ =4 · ξ2ω2

n

−4

ω2n

=4

ω2n

· (ξ2 − 1)

Δ ξ

ξ > 1 Δ > 0

r1 = ωn ·(−ξ−

√ξ2 − 1

)r2 = ωn ·

(−ξ+

√ξ2 − 1

)

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τ1 = −1

r1= −

1

ωn ·(−ξ−

√ξ2 − 1

) τ2 = −1

r2= −

1

ωn ·(−ξ+

√ξ2 − 1

)

ξ =τ1 + τ2

2 · √τ1 · τ2 ωn =1√

τ1 · τ2

H(p) =K

(1+ τ1 · p) · (1+ τ2 · p)

ξ = 1 Δ = 0

r = −ωn

H(p) =K(

1+p

ωn

)2=

K

(1+ τ · p)2

0 < ξ < 1 Δ < 0

r1 = ωn ·(−ξ− j ·

√1− ξ2

)r2 = ωn ·

(−ξ+ j ·

√1− ξ2

)

H(p) =K

1+2 · ξωn

· p+p2

ω2n

ξ

e(t) = E0 ·H(t) H(t)

S(t) =K

1+2 · ξωn

· p+p2

ω2n

· E0

p

ξ > 0 ωn > 0

t→∞ s(t) =p→0

p · S(p) =p→0

⎛⎜⎜⎝p · K

1+2 · ξωn

· p+p2

ω2n

· E0

p

⎞⎟⎟⎠

t→∞ s(t) = K · E0

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K

t→0s(t) =

p→∞p · S(p) =p→∞

⎛⎜⎜⎝p · K

1+2 · ξωn

· p+p2

ω2n

· E0

p

⎞⎟⎟⎠

t→0s(t) = 0

t→0s(t) =

p→∞p · p · S(p) =p→∞

⎛⎜⎜⎝p2 · K

1+2 · ξωn

· p+p2

ω2n

· E0

p

⎞⎟⎟⎠

t→0s(t) = 0

ξ ωn ξ > 0 ωn > 0

ξ > 1

S(t) =K

(1+ τ1 · p) · (1+ τ2 · p) ·E0

p

S(p) = K · E0 ·(1

p−

τ21τ1 − τ2

· 1

1+ τ1 · p −τ22

τ2 − τ1· 1

1+ τ2 · p)

s(t) = K · E0

⎛⎜⎝1−

τ1

τ1 − τ2· e

−t

τ1 −τ2

τ2 − τ1· e

−t

τ2 ·H(t)

⎞⎟⎠

ξ > 1 ξ =τ1 + τ2

2 · √τ1 · τ2ωn

z = 1

S(t) =K

(1+ τ · p)2 ·E0

p

S(p) = K · E0 ·(1

p−

τ

(1+ τ · p)2 −τ

(1+ τ · p))

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t

K · E00 95 ·K · E0

ξ > 1

ξ > 1

ξ = 1

s(t) = K · E0 ·⎛⎝1−

t+ τ

τ· e−

t

τ

⎞⎠ ·H(t)

ξ ωn

s(t) = K · (1− (1+ωn · t) · e−ωn·t) ·H(t)

0 < ξ < 1

S(p) =K

1+2 · ξωn

· p+p2

ω2n

· E0

p

s(t) = K · E0

(1−

1√1− ξ2

e−z·ωn·t ·(ωn

√1− ξ2 · t+ϕ

))·H(t) ϕ = ξ

ωp

ωp = ωn ·√1− ξ2

s(t) = K · E0

(1−

1√1− ξ2

e−zωnt · (ωp · t+ϕ)

)·H(t)

s(t) = K · E0

(1−

((ωp · t) + ξ√

1− ξ2· (ωp · t)

)· e−ξ·ωn·t

)·H(t)

Page 9: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

t

K · E00 95 ·K · E0

1 05 ·K · E0

0 < ξ < 1

0 < ξ < 1

ξ = 1

Tpm =Tp

2

Tp

t

K · E00 95 ·K · E0

1 05 ·K · E0

0 < ξ < 1

Tp =2 · π

ωn ·√1− ξ2

ωp = ωn ·√1− ξ2

Tpm =Tp

2=

π

ωn ·√

1− ξ2

D1 = e

−π · ξ√1− ξ2

Page 10: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

ξ ξ

D1

D2

D3

D4

D5

D6

D7

D8

D1

D2

D3D4

ξ

t

s

+5

−5

Tr0 7 Tr1

ξ = 1

ξ ≈ 0 7T5 ·ωn

ξ > 0 7

ξ

ξ > 0 7

ξ

ξ ≈ 0 7

ξ = 1

ξ = 0 7 D1 = e

−π · ξ√1− ξ2 = 0 05

T5

Page 11: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

z

Tr ·ωn

H(p) =K

1+ 2·zωn

p+ p2

ω2n

t

s

e(t) = E0 ·H(t)

s(t) = K · E0 · t ·H(t)

s(t)= K · e(t)

s(t) =

∫+∞0

K · e(u) u

H(p) =S(p)

E(p)=

K

pK

p

s(t) = K · e(t)

H(p) =S(p)

E(p)= K · p K · p

Page 12: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

s(t) = e(t− τ)

H(p) =S(p)

E(p)= e−τ·p

e−τ·p

−+

E(p)T(p)

ε(p) S(p)

BF(p) =S(p)

E(p)=

T(p)

1+ T(p)

T(p) =Ko

1+ τo · pKo τo

BF(p) =S(p)

E(p)=

Ko

1+ τo · p1+

Ko

1+ τo · p=

Ko

1+Ko + τo · p

BF(p) =

Ko

1+Ko

1+τo

1+Ko· p

Kf =Ko

1+Ko

τf =τo

1+Ko

Ko →∞

Page 13: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

e(t) = E0 ·H(t)

t→∞ s(t) =p→0

p · S(p) =p→0

(p · Kf

1+ τf · p ·E0

p

)= Kf · E0

t→∞ s(t) =Ko

1+Ko· E0

εi

εi =t→∞ ε(t) =

p→0p · ε(p) =

p→0(p · (E(p) − S(p)))

εi = (

(1−

Ko

1+Ko

)· E0 =

1

1+Ko· E0

Ko →∞

T(p) =Ko

1+2 · ξoωn0

· p+p2

ω2n0

Ko ξo ωno

BF(p) =

Ko

1+Ko

1+2ξ0

(1+Ko) ·ωno· p+

p2

(1+Ko) ·ω2no

Kf =K0

1+Ko

ωnf = ωno ·√1+Ko > ωno

ξf =ξo√1+Ko

< ξo

1 Ko

H(p) =K

1+ a1 · p+ a2 · p2 + · · ·+ an · pn

Page 14: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

H(p) =S(p)

E(p)=

1

(1+ p) ·(1+

p

6

)·(1+

p

20

)

s(t) = 1−25

19· e−t −

1

57· e−20·t +

1

3· e−5·t

e20·t

e−t e−5·t

H2(p) =1

(1+ p) ·(1+

p

6

) H1(p) =1

(1+ p)

t

s

H(p) H2(p)

t

s

H(p) H1(p)

Page 15: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

H(p) = K · 1+ a · p1+ b1 · p+ b2 · p2

H(p) = K ·(

1

1+ b1 · p+ b2 · p2+ a · p

1+ b1 · p+ b2 · p2

)

a

S(p) = K ·(

1

1+ b1 · p+ b2 · p2· 1p+ a · p

1+ b1 · p+ b2 · p2· 1p

)

s1(t) =−1

(1

1+ b1 · p+ b2 · p2· 1p

)

s(t) = K ·(s1(t) + a · s1(t)

)

t→∞ s(t) =p→0

p · S(p)

=p→0

(p ·K ·

(1

1+ b1 · p+ b2 · p2· 1p+ a · p

1+ b1 · p+ b2 · p2· 1p

))= K

s1(t)

0 5 · s1(t)

s(t)

t

s

H(p) =1+ 0 5 · p1+ p+ p2

=1

1+ p+ p2+

0 5 · p1+ p+ p2

s(t) = s1(t) + 0 5 · s1(t)

Page 16: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

H(p) =1− 0 5 · p1+ p+ p2

=1

1+ p+ p2+

0 5 · p1+ p+ p2

s(t)

s1(t)

0 5 · s1(t)

s(t)

t

s

ts

H(p) =K

(1+ τ · p) · (1− τ · p) τ > 0

e(t) = E0 ·H(t)

S(p) =K

(1+ τ · p) · (1− τ · p) ·E0

p

S(p) = K · E0 ·(1

p−

τ

2 · (1+ τ · p) +τ

2 · (1− τ · p))

s(t) = K · E0 ·⎛⎝1−

1

2

⎛⎝e

t

τ + e−t

τ

⎞⎠⎞⎠H(t)

−∞

Page 17: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

t

s

H(p) =1

1− p+ 10 · p2

Page 18: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

s(t)+ 10 · s(t) = 3 · u(t)

u(t) = K · (e(t) − s(t))

−+

E(p) ε(p) U(p) S(p)

K εi 5

e(t) = 2 ·H(t) 5

−+

E(p)A

ε(p)F(p)

U(p) S(p)

A A = 10

F(p)

e(t) = E0H(t) E0 = 5

0 08 s

F(p)F(p)

F(p) =K

1+ τ · p

T(p) =S(p)

E(p)K τ A

T(p) =G

1+ T · pG T

e(t) = E0H(t) E(p) e(t) S(p)G

Page 19: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

s(t) T

s(t) s(t)

A = 10 τ K F(p)

A

t

s

e(t)

200mm 1200mm

y0

y0

y(t)

yc(t)

Page 20: +@ UCja3 úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b · +@ UCja3 8úim/2i2KTQ`2HH2/2b bvbi K2bHBMû B`2b 9YSavbi K2b/mR2`Q`/`2 9YSYS.û}MBiBQM mNcwcj L30nUa3LC3aRa0a33cj0 ,aCjU anN3 \n

εi = 0

εi =t→∞ (y(t) − yc(t))

T5 � 5 s

d1 < 5

d1 =ymax − y∞

y∞

Hm(p) =Y(p)

M(p)=

Km

1+ a1 · p+ a2 · p2

qm(t) =m(t)

a1 = 0 7 s a2 = 45 9× 10−3 s−2 Km = 2mkg−1

qm(t) ur(t)

Hd(p) =Qm(p)

Ur(p)= Kd = 0 2 kg s−1 V−1

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um(t) y(t)ε(t) = uc(t) −um(t) A

uc(t) yc(t)

Hc(p) =Um(p)

Y(p)= Kc = 10V m−1

Uc(p) = G · Yc(p)

GYc(p)

−+

Uc(p)A

ε(p)Hd(p)

Ur(p)Hv(p)

Qm(p)Hm(p)

M(p) Y(p)

Hc(p)

Um(p)

G = Kc Hv(p) =1

p

HBO(p) =Um(p)

ε(p)

HBO(p) HBO(p) =K

p ·(1+ 2 · ξ

ω0· p+

p2

ω20

)K ξ ω0

HBF(p) =Y(p)

Yc(p)

HBF(p) =KBF

1+ b1 · p+ b2 · p2 + b3 · p3