APPENDICE A : lista dei comandi Mathcad per la progettazione di dettaglio dei rulli di sostegno e movimentazione della lastra

(1)

APPENDICE A

APPENDICE A : lista dei comandi Mathcad per la progettazione di

dettaglio dei rulli di sostegno e movimentazione della lastra

s

3 d3

MI

D2

1 d1

2 QI

LI

NI

RA

RB

Inserire il carico sul rullo [Kg]

Ql:=

7000

⋅kg

Inserire la lunghezza del tratto Ml [mm]

Ml:=

145

⋅mm

Inserire la lunghezza del tratto Ll [mm]

Ll:=

2200

⋅mm

Inserire la lunghezza del tratto Nl [mm]

Nl:=

145

⋅mm

Inserire il valore di E1 [N/mm^2}

E1

220000

N mm

2

⋅ :=

Inserire il valore di E2 [N/mm^2}

E2

220000

N mm

2

⋅ :=

Inserire il valore di E3 [N/mm^2}

E3

220000

N mm

2

⋅ :=

(2)

APPENDICE A

Inserire il valore del diametro d1 [mm]

d1:=

50

⋅mm

Inserire il valore dello spessore di mantello s [mm]

s:=

20

⋅mm

Inserire il valore del diam.esterno mantello D2 [mm]

D2:=

220

⋅mm d2 D2:= −

2

⋅s

Inserire il valore del diametro d3 [mm]

d3:=

50

⋅mm

Inserire densita' tratto M [Kg/mm^3]

dens1

7.8 10

⋅ −

6

kg mm

3

⋅ :=

Inserire densita' tratto M [Kg/mm^3]

dens2

7.8 10

⋅ −

6

kg mm

3

⋅ :=

Inserire densita' tratto M [Kg/mm^3]

dens3

7.8 10

⋅ −

6

kg mm

3

⋅ :=

Peso totale del rullo [N]

P π d1

2

4

⋅ ⋅Ml⋅_dens1⋅g π d3

2

4

⋅ ⋅Nl⋅_dens3⋅g + π D2

2

d2

2

−









4

⋅ _{⋅ dens3}Ll⋅ ⋅g + := Massa_rullo P g := Massa_rullo =

220.08kg

Qlf Ql g⋅:=

(3)

APPENDICE A

Qc

2

3

⋅Qlf Ll := Q _Qc P Ll + := Ra Q Ll⋅ Nl Ll

2

+











Ll+Ml+Nl ⋅ := Ra=

2.396 10

×

4 N

Rb Q Ll⋅ Ml Ll

2

+











Ll+Ml+Nl ⋅ := Rb=

2.396 10

×

4 N

S1 π d1

2

4

⋅ := S1=

1.963 10

× −

3 m

2 S2 π D2

2

d2

2

−









4

⋅ := S2=

0.013m

2 S3 π d3

2

4

⋅ := S3=

1.963 10

× −

3 m

2 J1 π d1

4

64

⋅ := J1=

3.068 10

× −

7 m

4

(4)

APPENDICE A

J2 π D2

4

d2

4

−









64

⋅ := J2=

6.346 10

× −

5 m

4 J3 π d3

4

64

⋅ := J3=

3.068 10

× −

7 m

4 A

0

Ml

1

mm ⋅

0

1

0

1

0

1

0

Ml −

1

mm ⋅ Ml+Ll ( )

1

mm ⋅

1

−

1

0

1

−

1

0

Ll+Ml+Nl ( )

1

mm ⋅

0

Ml − −Ll ( )

1

mm ⋅

0

1

−

0

1

0

1

−

0

0 











:= A−

1

4.016

− ×

10

−

4

1

4.016

− ×

10

−

4

1

4.016

− ×

10

−

4

1 4.016 10

× −

4

0 4.016 10

× −

4

0 4.016 10

× −

4

0 4.016 10

× −

4

0 4.016 10

× −

4

1

−

4.016 10

× −

4

1

−

4.016 10

× −

4

0 4.016 10

× −

4

0 4.016 10

× −

4

1

−

0.942

0

0.058

−

145

0.058

−

145

0.058

0

0.058

0

0.942

−

2.345 10

×

3 











= A1:=

0

A2





_{E3 J3}

1

_⋅







Ra Q Ll− ⋅

(

)

⋅(Ll+Ml+Nl)

3

6

Q Ll⋅ Ml⋅ ⋅(Ll+Ml+Nl)

2

+ Q Ll

2

⋅ ⋅(Ll+Ml+Nl)

2

4

+













⋅ := A3 Q Ml

4

E2 J2⋅ ⋅

8

⋅











Ra Ml⋅

3

6 









1

E1 J1⋅

1

E2 J2⋅ −











⋅ + := A4





−Ra_{E3 J3}+_⋅Q Ll⋅ +_{E2 J2}Ra_⋅







Ll+Ml ( )

3

6

⋅













Q Ll⋅ Ml⋅ ⋅(Ml+Ll)

2

⋅_E3⋅_J3 − Q Ll

2

⋅ ⋅(Ml+Ll)

2

E3 J3⋅ ⋅

4

− Q Ll( +Ml)

4

⋅

24

⋅_E2⋅_J2 − Q Ml⋅ (Ml+Ll)

3

⋅

6

⋅_E2⋅_J2 + Q Ml

2

⋅ ⋅(Ml+Ll)

2

4

⋅_E2⋅_J2 − :=

(5)

APPENDICE A

A5



_

_{E1 J1}

1

_⋅ − _{E2 J2}

1

_⋅







⋅Ra Ml

2

⋅













Q Ml⋅

3

6

⋅_E2⋅_J2 + := A6 Ra − +Q Ll⋅ E3 J3⋅ Ra E2 J2⋅ +











Ll+Ml ( )

2

⋅













Q Ll⋅ Ml⋅ ⋅(Ml+Ll) E3 J3⋅ − Q Ll

2

⋅ ⋅(Ml+Ll) E3 J3⋅ ⋅

2

− Q Ll( +Ml)

3

⋅

6

⋅_E2⋅_J2 − Q Ml⋅ (Ml+Ll)

2

⋅

2

⋅_E2⋅_J2 + Q Ml

2

⋅ ⋅(Ml+Ll)

2

⋅_E2⋅_J2 − := C A−

1

A1 A2⋅_mm

1

A3⋅_mm

1

A4⋅_mm

1

A5 A6













⋅ := C

4.698 10

× −

3

0 9.832 10

× −

4

0.359

1.096

901.752

−













=

Freccia tratto 1

f1 x( ) −Rax

3

⋅

6

⋅_E1⋅_J1 mm

2

⋅ C

0 0

, ⋅x + C

1 0

, +











mm ⋅ :=

Freccia tratto 2

f2 x( ) −

1

E2 J2⋅ Ra x⋅

3

6

mm

2

⋅ Q x

4

24

⋅ ⋅mm

3

− Q Ml⋅ x

3

6

⋅ ⋅mm

2

+ Q Ml⋅

2

x

2

4

⋅ ⋅mm −











⋅ C

2 0

, ⋅x + C

3 0

, +













⋅mm :=

Freccia tratto 3

f3 x( ) −

1

E3 J3⋅ Ra Q Ll− ⋅

(

)

⋅x

3

6

mm

2

⋅ Q Ll⋅ Ml⋅ x

2

⋅ ⋅mm + Q Ll⋅

2

x

2

4

⋅ ⋅mm +













⋅ C

4 0

, ⋅x + C

5 0

, +













⋅mm :=

(6)

APPENDICE A

Rotazione tratto 1

r1 x( ) Ra − ⋅x

2

⋅_E1⋅_J1 mm

2

⋅ C

0 0

, + :=

Rotazione tratto 2

r2 x( ) −

1

E2 J2⋅ Rax

2

mm

2

⋅ Q x

3

6

⋅ ⋅mm

3

− Q Ml⋅ x

2

⋅ ⋅mm

2

+ Q Ml⋅

2

x

2

⋅ ⋅mm −











⋅ C

2 0

, + :=

Rotazione tratto 3

r3 x( ) −

1

E3 J3⋅

(

Ra Q Ll− ⋅

)

x

2

⋅ ⋅mm

2

+Q Ll⋅ Ml⋅ ⋅ mmx⋅ Q Ll⋅

2

x

2

⋅ ⋅mm +













⋅ C

4 0

, + :=

Momento

M1 x( ):=_{Ra x⋅ mm}⋅ M2 x( ) _{Ra x⋅ mm}⋅ Q x mm( ⋅ −Ml)

2

⋅

2

− := M3 x( ) _{Ra x⋅ mm}⋅ Q Ll⋅ x mm⋅ −Ml Ll

2

−











⋅ − := Mt x( ) _{M1 x}( )

(

x≥

0 )

x Ml

1

mm ⋅ ≤











∧ if M2 x( ) x Ml

1

mm ⋅ >











x (Ml+Ll)

1

mm ⋅ ≤













∧ if M3 x( ) x (Ml+Ll)

1

mm ⋅ >













x (Ml+Ll+Nl)

1

mm ⋅ ≤













∧ if :=

(7)

APPENDICE A

0

500 1000

1500

2000

2

.

10

4

1

.

₁₀

4

0

lunghezza del rullo in mm

valore assunto dal momento in N

xm Mt x( ) − x

Taglio

Tt x( ) xMt x( )

1

mm ⋅ d d :=

0

500 1000

1500

2000

2

.

₁₀

4

0

2

.

10

4

valore assunto dal taglio in N

Tt x( )

(8)

APPENDICE A

Tensione

σfl x( ) Mt x( ) d1

2

⋅











J1

(

x≥

0 )

x Ml

1

mm ⋅ ≤











∧ if Mt x( ) D2

2

⋅











J2 x Ml

1

mm ⋅ >











x (Ml+Ll)

1

mm ⋅ < ∧ if Mt x( ) d3

2

⋅











J3 x (Ml+Ll)

1

mm ⋅ ≥ x (Ml+Ll+Nl)

1

mm ⋅ ≤













∧ if :=

0

500 1000

1500

2000

0

1

.

10

8

2

.

₁₀

8

3

.

10

8 σfl x( ) x

τ di taglio

τ x( ) Tt x ( )

(

)

S1

(

x≥

0 )

x Ml

1

mm ⋅ ≤











∧ if Tt x( )

(

)

S2 x Ml

1

mm ⋅ >











x (Ml+Ll)

1

mm ⋅ <













∧ if Tt x( )

(

)

S3 x (Ml+Ll)

1

mm ⋅ ≥













x (Ml+Ll+Nl)

1

mm ⋅ ≤













∧ if :=

(9)

APPENDICE A

0

500 1000

1500

2000

1

.

₁₀

7

0

1

.

₁₀

7 τ x( ) x

Tensione equivalente di Von Mises

σwm x( ):=



_

(

_{σfl x}( )

)

2

+

3

⋅(τ x( ))

2 

_

0

500 1000

1500

2000

0

1

.

10

8

2

.

10

8

3

.

₁₀

8

tensione equivalente di von mises

σwm x( ) x

Freccia

ft x( ) _{f1 x}( )

(

x≥

0 )

x Ml

1

mm ⋅ ≤











∧ if f2 x( ) x Ml

1

mm ⋅ >











x (Ml+Ll)

1

mm <













∧ if f3 x( ) x (Ml+Ll)

1

mm ⋅ ≥













x (Ml+Ll+Nl)

1

mm ⋅ ≤ ∧ if :=

(10)

APPENDICE A

0

500 1000

1500

2000

0.002

0.001

0

freccia in mm

ft x( ) −

x

Valore di massima tensione sul tampone-perno 1

x:=

0.1

Given x≥

0

x (Ml)

1

mm ⋅ ≤ x__{σmax1 Maximize σwm x}:=

(

,

)

x__σmax1=

145

σwm x_σmax1

(

)

=

2.839 10

×

8 Pa

σmax1 σwm x_σmax1:=

(

)

La tensione massima sul tampone-perno1 e' [N/mm^2]

σmax1 =

2.839 10

×

8 Pa

x__{σmax1 mm}⋅ =

0.145m

Valore di massima tensione sul tampone-perno 2

x (Ll+Ml)

1

mm ⋅ := Given x (Ll+Ml)

1

mm ⋅ ≥ x (Ll+Ml+Nl)

1

mm ⋅ ≤

(11)

APPENDICE A

x__{σmax2 Maximize σwm x}:=

(

,

)

x__σmax2=

2.345 10

×

3

σwm x_σmax2

(

)

=

2.839 10

×

8 Pa

σmax2 σwm x_σmax2:=

(

)

La tensione massima sul tampone-perno2 e' [N/mm^2]

σmax2 =

2.839 10

×

8 Pa

x__{σmax2 mm}⋅ =

2.345m

Valore di massima tensione sul mantello

x Ml

1

mm ⋅ := Given x Ml

1

mm ⋅ +

1

> x (Ll+Ml)

1

mm ⋅ −

1

< x__{σmax Maximize σwm x}:=

(

,

)

x__σmax=

1.245 10

×

3

σwm x_σmax

(

)

=

2.887 10

×

7 Pa

σmax σwm x_σmax:=

(

)

La tensione massima sul mantello e' [N/mm^2]

σmax=

2.887 10

×

7 Pa

x__{σmax mm}⋅ =

1.245m

x Ll

1

mm ⋅ := Given x≥

0

x (Ll+Ml+Nl)

1

mm ⋅ ≤ x_fmax Maximize ft x:=

( )

,

(12)

APPENDICE A

x_fmax=

1.245 10

×

3

ft x_fmax

(

)

=

1.127 10

× −

3 m

fmax ft x_fmax:=

(

)

La freccia massima del rullo e' [mm]

fmax=

1.127 10

× −

3 m

Vf:= _Ll₊fmax_Ml₊_Nl

Vf=

4.527 10

× −

4

V1:=

"L' albero non e' verificato a rigidezza"

V2:=

"L' albero e' verificato a rigidezza per applicazioni di scarsa importanza "

V3:=

"L' albero e' verificato a rigidezza per applicazioni: comuni e di scarsa importanza "

V4:=

"L' albero e' verificato a rigidezza per applicazioni: su riduttori,comuni e di scarsa importanza "

V5:=

"L' albero e' verificato a rigidezza per applicazioni: su macchine utensili, su riduttori, comuni e di scarsa importanza "

V _V1 if _Vf>

0.001

V2 if Vf≤

0.001

∧_Vf>

0.0005

V3 if Vf≤

0.0005

∧_Vf>

3.33310

⋅ −

4

V4 if Vf≤

2.5 10

⋅ −

4

∧_Vf>

2 10

⋅ −

4

V5 if Vf≤

2 10

⋅ −

4

:=