Costruzione di macchine 2 Esercizi

Capitolo 4

Esercizio 4.1

r_i = 180 mm

r_e = 220 mm

p_i = 100 MPa

p_e = 40 MPa

σ_T = A - B^r_i/_r² + J_e(1-ν)

σ_E = A + B^r_i/_r² + J_e

J_e = p_ir_i² - p_er_e²/_{re² - ri²}

σ_r = r_i

ρ = r_e

σ_r = A - r_e² B_i/_r² + p_ir_i² - p_er_e² 1-ν = -p_ir_i

ρ = r_e

σ_r = A - B p_ir_i² - p_er_e² 1-ν = -p_e

p_ir_i² - p_er_e²/_re² - 1-ν = -p_e + B - A

A = p_e² B + p_e + B - A = -p_e

^{p_i2/_r2 B} = (B/p) = p_e - p_i

B = (p / p_e) B - B / a²

A = p_e (p - B) r_e² c_i² = p_i r_i² - B / 1-ν

σ_r = ^{p_ir_i2 - p_er_e2}/_{re² - ri²} - ^r_e2/_r²p_e

σ_t = ^{p_ir_i2 - p_er_e2}/_{re² - ri²} + ^r_e2/_r²p_e

σ_E = ^p_ir_i2/_{re² - ri²} = σ_E

σ_t + σ_E = ^{p_ir_i2 - p_er_e2}/_{re² - ri²} = σ_E

- punto A r = r_e

σ_r = ^{p_ir_i2 - p_e}/_{re² - ri²}r_e² - ^p_i/_r²r_i² = -40 MPa

σ_t = ^p_ir_i2/_{re² - ri²}r_e² + ^r_i/_re²p_er² = 203 MPa

σ_E = 81,5 MPa

J_idvn = ^{√(σ_r2 + σ_t2 + σ_E2 - (σ_rσ_t + σ_rσ_E + σ_tσ_E))} = 210,444 MPa

punto B r = r_i

σ_r = ^{p_ir_i2 - p_e}/_{re² - ri²}r_e² - ^p_i/_r²p_e = -100 MPa

σ_t = ^p_ir_i2/_{re² - ri²}r_e² + ^r_e2/_ri²p_er² = 263 MPa

σ_E = 81,5 MPa

J_idvn = 314,367 MPa

Disco 1

σ_r1 = 0

σ_t1/ri = 123,89 MPa

σ_r1au = 37,92 MPa

σ_t1au = 52,46 MPa

Disco 2

σ_r2/ri = 75,836 MPa

σ_t2/ri = 63,84 MPa

σ_r2/re = 0

σ_t2/re = 32,92 MPa

Esercizio 4.5

r_i = 20 mm r_uo = 100 mm r_e = 200 mm b₁ = 40 mm b₂ = 20 mm n = 10000 giri/min

r =          c = r_e⁴₃γ 3+2   ω²r_e²      (2π 10000)² C =    σ_o = 7850 – 6970 8       60   0,2 · 10^-6 = 344,34 MPa

D = 142,04 MPa 3 3 (1+3)D       σ_o = 81,78 MPa

Disco 1 σ_r1 = A₁ - B₁ - C σ_t1 = A₁ + B₁ - D

Disco 2 σ_r2 = A₂ - B₂ - C σ_t2 = A₂ + B₂ - D

Condizioni al contorno: σ_r1|_r=1 = 0 σ_r2|_r=e 0 σ_t1|₁ = σ_r2|_b2 = r = r_um r|_r=um = σ_t2|_r=um

Condizioni:

• A₁ - 100B₁ - 0,01C = 0
• A₂ - B₂ - C = 0
• (A₁ - 0,4B₁ - 0,25C)40 = (A₂ - 4B₂ - 0,25(1)20
• A₁ + 4B - 0,3A₁ + 1,2B - A₂ + 8A₂+ 0,3A₂ - 1,2B₁
• 0,7A₁ + 5,2B₂ - 0,7A₂ - 5,2B₂

[matr. x]₂

[1 -100 0 1] [0 0 1 -1] [40 -160 -20 -80] [2,7 5,2 -0,7 5,2]

σ_re = _pcr²/_r²-r²e * (1 - _r²e/_r² ) = 0

σ_te = _pcr²/_r²-r²e * (1 + _r²e/_r² ) = 38,02 MPa

r = r_i

σ_ri = -15 MPa

σ_ti = -55,02 MPa

σ_tide = 55,02 MPa

r = r_o

σ_ro = -23,76 MPa

σ_to = 46,21 MPa

σ_tideo = 46,27 MPa

r = r_c

σ_rc = -23,76 MPa

σ_tc = 70,33 MPa

σ_tidec = -94,01 MPa

b = r_c

σ_rc = 0 MPa

σ_tc = 54,9 MPa

σ_tidec = 54,9 MPa

b³ = 2,93

Esercizio 4.15

f_e = 200 mm r_o = 150 mm r_i = 100 mm E = 206000 MPa Φ = 0,25 mm σ_S = 240 MPa μ = 0,45 t = 10 mm

J_U = 27,82 MPa

Μ_Umax = 23 468,40 N·m

Disco 1

p_E

• σ_T = -35 MPa
• σ_TE = 58,33 MPa
• σ_rc = -9,07 MPa
• σ_TE = 32,41 MPa
• σ_rc = 0
• σ_TE = 23,3 MPa

Condizione p_C, recipiente interno

• σ_rc = 0
• σ_TE = -100,15 MPa
• σ_rc = -27,82 MPa
• σ_TE = -72,33 MPa

Intervallo

• • a': 0,004
  • α(a'): 1
  • a'': 0,034
  • α(a''): 1,014321
  • x̅: 1,0071605
• • a': 0,034
  • α(a'): 1,014321
  • a'': 0,064
  • α(a''): 1,053021
  • x̅: 1,033671
• • a': 0,064
  • α(a'): 1,053021
  • a'': 0,094
  • α(a''): 1,123461
  • x̅: 1,088241
• • a': 0,094
  • α(a'): 1,123461
  • a'': 0,115
  • α(a''): 1,199815
  • x̅: 1,161430

N^r' = 488322,0003 cicli

N^r'' = 63937,823 cicli

V^r' = 25757,149 cicli

V^r'' = 9581,157 cicli

l_tot = 587598,129 cicli