Erzwungene Schwingungen. Hookesches Gesetz.

Erzwungene SchwingungenErzwungene Schwingungen

F = - Ds - bdsdt + F a


F = - Ds - bdsdt + F a cos t


s = s cos(t - 0)F = - Ds - bdsdt + F a cos t




s() = Fa/m

(20-2)2 + (2)2

s = s cos(t - 0)



20 = D/m

s() = Fa/m

(20-2)2 + (2)2

s = s cos(t - 0)



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

s = s cos(t - 0)



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s = s cos(t - 0)



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

1) 0



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

1) 0 F a/m20



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

1) 0 F a/m20

Hookesches Gesetz



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

s

00

1

1) 0 F a/m20

Hookesches Gesetz



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

s

1

1) 0 F a/m20

2) 0

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

1) 0 F a/m20

2) 0 F a/0b

s

1

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

1) 0 F a/m20

2) 0 F a/0b

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

1) 0 F a/m20

2) 0 F a/0b

r = 2

0 - 22

r

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

1) 0 F a/m20

2) 0 F a/0b

r = 2

0 - 22

r

Resonanzfrequenz

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

r

1) 0 F a/m20

2) 0 F a/0b

3)

r = 2

0 - 22

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

r = 2

0 - 22

r

1) 0 F a/m20

2) 0 F a/0b

3) 0

3

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

r = 2

0 - 22

r

3

1) 0 F a/m20 0

2) 0 F a/0b

3) 0

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

r = 2

0 - 22

r

3

1) 0 F a/m20 0

2) 0 F a/0b

3) 0

0

0

1

0

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

r = 2

0 - 22

r

3

0

0

1 2

1) 0 F a/m20 0

2) 0 F a/0b /2

3) 0

0

0

/2

00



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

s = s cos(t - 0)

0

s

1

2

r = 2

0 - 22

r

3

0

/2

1 2

1) 0 F a/m20 0

2) 0 F a/0b /2

3) 0

3

0

00

00

SW04.1 Resonanzkurven.nb



= b/2m

20 = D/m

s() = Fa/m

(20-2)2 + (2)2

() = arctan

22

0-2

s 0

0

s

1

2

r = 2

0 - 22

r

3

0

/2

1 2

1) 0 F a/m20 0

2) 0 F a/0b /2

3) 0

3

0

s = s cos(t - 0)

00

00

Fa = F a cos t s = s cos(t - 0)

s = s cos(t - 0)

0 = 0

Fa = F a cos t

Fa

s = s cos(t - 0)

0 = 0

Fa = F a cos t

Fa

s

s = s cos(t - 0)

0 = 0

Fa = F a cos t

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

I II III IV

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

Fa

s

s.

s = s cos(t - 0)

s.Fa

0 = 0

Fa = F a cos t

I II III IV

Fa

s

s.

s = s cos(t - 0)

s.Fa - + - +

0 = 0

Fa = F a cos t

I II III IV

0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

I II III IV

s.Fa - + - +

0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

s.Fa - + - +

I II III IV I II III IV

0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

s.Fa - + - +


0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

s.Fa - + - +


+ + + +

0 =

0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

s.Fa - + - +


+ + + +

I II III IV

0 =

0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

s.Fa - + - +


+ + + +

I II III IV

0 =

0 = /2

Fa

s

s.

s = s cos(t - 0)

0 = 0

Fa = F a cos t

s.Fa - + - +


+ + + +

I II III IV

+ - + -

Resonanzkurve

s

0

0

Resonanzkurve

Gütefaktor Q s

Q r/2

r

00

Resonanzkurve

Gütefaktor Q s

Q r/2

r

00

Halbwertsbreite 2

Resonanzkurve

Gütefaktor Q s

Q r/2

r

00

71%

s (r)

s (r)/ 2

Resonanzkurve

Gütefaktor Q s

Q r/2

r

00

Q s (r)

s (0)

s (r)

Resonanzkurve

Gütefaktor Q

Q r/2

Q s (r)

s (0)

Q 2 WW

s

0

0

Resonanzkurve

Gütefaktor Q

Q r/2

Q s (r)

s (0)

Q 2 WW

s

0

0 W = D 2s (r)

12

Gütefaktor Q

Q r/2

Q s (r)

s (0)

Q 2 WW

W = D 2s (r)12

s

0

0

2=

Gütefaktor Q

Q r/2

Q s (r)

s (0)

Q 2 WW

W = D 2s (r)12

s

0

0 =f

2==f

Gütefaktor Q

Q r/2

Q s (r)

s (0)

Q 2 WW

W = D 2s (r)12

s

00 f

f =/

s

0

0

SW04.1 Resonanzkurven.nb

[2.28] Die Schwingungsamplituden eines Turbinengehäuses wurden beiverschiedenen Drehzahlen (=Frequenzen) gemessen:

Amplitude/mm 2,5 15 37(Max)

20

Drehzahl/Upm 100 1650 1800 1900

Die effektive Masse des Gehäuses beträgt 2,3 t.

Bestimmen Sie die Resonanzfrequenz fr, die Abklingkonstante , dieDämpfungskonstante b, die Eigenfrequenz f0, die effektive Federkonstante D undden Gütefaktor Q. (Resonanzkurve durch ein Dreieck annähern.)

Wieviel Energie W steckt bei Resonanz im Gehäuse? Wieviel Leistung P wird inWärme umgewandelt?

Da die Turbine überwiegend bei 2000 Upm betrieben werden soll, muß dieResonanzfrequenz auf fneu < 1500 min-1 vermindert werden. Wie kann dasgeschehen?

Erzwungene Schwingungen. Hookesches Gesetz.

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