p1 := 1000 ⋅ kN m1 p1

(1)

VALIDAZIONE SAP ANALISI STEADY STATE (ing S. Caffè): ORIGIN := 1

Masse nodali:

p1 := 1000 ⋅ kN m1 p1

g

101971.621 kg

= :=

p2 := 5000 ⋅ kN m2 p2

g

509858.106 kg

= :=

Μ m1

0 0

m2



 



 

1 ⋅ kg :=

Rigidezze e smorzamenti viscosi dei link:

k1 1000 kN

⋅ m

:= c1 10 kN s ⋅

⋅ m :=

k2 500 kN

⋅ m

:= c2 50 kN s ⋅

⋅ m :=

Κ k1 k2 +

− k2

− k2 k2



 



 

m

⋅ N 1500000 500000

−

500000

− 500000

 



 

= 

:=

CVL c1 c2 +

− c2

− c2 c2



 



 

m N s ⋅

⋅ 60000

50000

−

50000

− 50000

 



 

= 

:=

Frequenze proprie del sistema:

Λ sort eigenvals Μ ( ( ⁻ ¹ ⋅ Κ ) ) ^0.6389

15.0517

 



 

= 

:=

ωn1 Λ

1 = 0.799

:= fn1 ωn1

2 ⋅ π = 0.1272 :=

ωn2 Λ

2 = 3.88

:= fn2 ωn2

2 ⋅ π = 0.617 :=

Forzanti armoniche:

(2)

FDYN f() F1

N 2 ⋅ π ⋅ f ( ) ²

2 ⋅ π ⋅ f1

( ) ²

⋅

F2 N

2 ⋅ π ⋅ f ( ) ²

2 ⋅ π ⋅ f2

( ) ²

⋅

 



 

 



 

:=

Integrazione diretta delle equazioni del moto:

KDYN f()  Κ + − 1 ⋅ ( 2 ⋅ π ⋅ f ) ⋅ CVL − ( 2 ⋅ π ⋅ f ) ² ⋅ Μ

 

:= 

U f ( ) := KDYN f() ⁻ ¹ ⋅ FDYN f() u1 f() U f ( )

:= 1 u2 f() U f ( )

:= 2

f := 0 0.01 , .. 3

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3

0 2 10 ×

⁻⁵

4 10 ×

⁻⁵

6 10 ×

⁻⁵

8 10 ×

⁻⁵

1 10 ×

⁻⁴

1.2 10 ×

⁻⁴

1.4 10 ×

⁻⁴

1.6 10 ×

⁻⁴

1.8 10 ×

⁻⁴

2 10 ×

⁻⁴

u1 f() u2 f()

f

(3)

Eliminazione smorzamento link e introduzione fattori "alfa" e "beta":

α := 0 β := 0.06

Smorzamento Isteretico CH := ( α Μ ⋅ + β Κ ⋅ )

KDYN_bis f()  Κ + − 1 ⋅ CH − ( 2 ⋅ π ⋅ f ) ² ⋅ Μ

 

:= 

Ubis f() := KDYN_bis f() ⁻ ¹ ⋅ FDYN f() u1_bis f() := Ubis f() ₁

u2_bis f() := Ubis f() ₂ , ..

:=

(4)

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 0

8 10 ×

⁻⁵

1.6 10 ×

⁻⁴

2.4 10 ×

⁻⁴

3.2 10 ×

⁻⁴

4 10 ×

⁻⁴

4.8 10 ×

⁻⁴

5.6 10 ×

⁻⁴

6.4 10 ×

⁻⁴

7.2 10 ×

⁻⁴

8 10 ×

⁻⁴

u1_bis f() u2_bis f()

f

(5)

Smorzamento link e introduzione fattori "alfa" e "beta":

α = 0 β = 0.06 CH

90000 30000

−

30000

− 30000

 



 

=  CVL

60000 50000

−

50000

− 50000

 



 

= 

CTOT f() CH 2 ⋅ π ⋅ f + CVL :=

KDYN_ter f()  Κ + − 1 ⋅ ( 2 ⋅ π ⋅ f ) ⋅ CTOT f() − ( 2 ⋅ π ⋅ f ) ² ⋅ Μ

p1 := 1000 ⋅ kN m1 p1

VALIDAZIONE SAP ANALISI STEADY STATE (ing S. Caffè): ORIGIN := 1

Masse nodali:

p1 := 1000 ⋅ kN m1 p1

g

101971.621 kg

= :=

p2 := 5000 ⋅ kN m2 p2

g

509858.106 kg

= :=

Μ m1

0 0

m2



 



 

1

⋅ kg :=

Rigidezze e smorzamenti viscosi dei link:

k1 1000 kN

⋅ m

:= c1 10 kN s ⋅

⋅ m :=

k2 500 kN

⋅ m

:= c2 50 kN s ⋅

⋅ m :=

Κ k1 k2 +

− k2

− k2 k2



 



 

m

⋅ N 1500000 500000

−

500000

− 500000

 



 

= 

:=

CVL c1 c2 +

− c2

− c2 c2



 



 

m N s ⋅

⋅ 60000

50000

−

50000

− 50000

 



 

= 

:=

Frequenze proprie del sistema:

Λ sort eigenvals Μ ( ( − 1 ⋅ Κ ) ) 0.6389

15.0517

 



 

= 

:=

ωn1 Λ

1 = 0.799

:= fn1 ωn1

2 ⋅ π = 0.1272 :=

ωn2 Λ

2 = 3.88

:= fn2 ωn2

2 ⋅ π = 0.617 :=

Λ sort eigenvals Μ ( ( ⁻ ¹ ⋅ Κ ) ) ^0.6389

N 2 ⋅ π ⋅ f ( ) ²

( ) ²

2 ⋅ π ⋅ f ( ) ²

( ) ²

KDYN f()  Κ + − 1 ⋅ ( 2 ⋅ π ⋅ f ) ⋅ CVL − ( 2 ⋅ π ⋅ f ) ² ⋅ Μ

U f ( ) := KDYN f() ⁻ ¹ ⋅ FDYN f() u1 f() U f ( )

KDYN_bis f()  Κ + − 1 ⋅ CH − ( 2 ⋅ π ⋅ f ) ² ⋅ Μ

Ubis f() := KDYN_bis f() ⁻ ¹ ⋅ FDYN f() u1_bis f() := Ubis f() ₁

u2_bis f() := Ubis f() ₂ , ..