shaft design

Shaft Design ENTC 463 Mechanical Design Applications II

Next Thursday 4/3/2008 Meet @ Thompson 009B  Allen to Hagan – 2:20 to 3:00 PM Higginbotha Higginbotham m to to Winnifo Winniford– rd– 3:00 to 3:35 3:35 PM

ENTC 463 Mechanical Design Applications II

Next Thursday 4/3/2008 Meet @ Thompson 009B  Allen to Hagan – 2:20 to 3:00 PM Higginbotha Higginbotham m to to Winnifo Winniford– rd– 3:00 to 3:35 3:35 PM

ENTC 463 Mechanical Design Applications II

ATTENTION: MMET Students, the IAC members would like to meet YOU on April 11th ! WHO? IAC Members WHAT? Meet and Greet WHERE? 510, 5th Floor Rudder Tower WHEN? April 11th 3:30-5:00pm WHY? They will also be available to review your resume and help you improve it for potential employers. Please sign up with Courtney in THOM 117

Deadline to sign up is April 8th.

MMET IAC Meeting April 11th 2008

STUDENTS NEEDED!! MMET Majors ONLY Come join us for the 2008 Spring IAC Meeting Great DOOR PRIZES! I-POD NANO and much more! (Must Be Present to Win)

Sign up with Courtney in THOM 117

ENTC 463 • Lab 3 Due 4/10 • Homework HW 12 Due 4/15  – Chapter 12 – 2, 4, 25, 27

Shafts • Mott, Chapter 12 • Why use shaft?  – To transmit power

• Shaft geometry  – Cylinder, bar, beam (length and diameter)

• Load acting on shaft  – Torsion (shear stress)  – Bending (normal stress)

Shaft Design

Shaft Design

Shaft Design • Given required power to be transmitted  – Calculate torque,  – Calculate forces,  – Calculate stresses (if geometry is known),  – Select material

• Given required power to be transmitted  – Calculate torque,  – Calculate forces,  – Determine shaft diameter (if the material is known)

Shaft Design Procedure 1. Develop the free body diagram; model the various machine elements mounted on the shaft in terms of forces and torques 2. Develop the shear and moment diagram; identify bending moment (leads to normal stress) and torque (leads to shear stress) 3. Identify critical locations for stress analysis; calculate stresses (known diameter) 4. Determine diameter or select material based on failure theories

Forces Acting on Shaft • Forces due to gear (spur gear) T  =

63000hp

W t  =

n T   D 2

W r  = W t  tan φ  W  x = W t  tanψ  (helical gear)

Forces due to Gears

Forces Acting on Shaft • Forces due to chain and sprocket  F c =

T   D 2

=

T  A  D A 2

 F cx =  F c cos θ   F cy =  F c sin θ 

=

T  B  D B 2

Forces Acting on Shaft • Forces due to V-belt and sheave  F  B =  F 1 − F 2  F  B ≈

1.5T 

 D 2

For flat belt and pulley

 F  B ≈

2.0T 

 D 2

Example • A chain is transmitting 100 kW with the chain speed at 6000 rpm and V = 50 m/s. The shaft material is AISI 1040 cold drawn. Determine the shaft diameter required.

Shaft Design Considerations • Stress Concentration (fillet or key seat) 1.5 < Kt < 2.5

• Combined tangential and radial load (3-D)  – Two shear and moment diagrams Wt

z

 M  y =  M  xy + M  2

Wr x

y

2  yz 

Stress Concentration • Keyseats  – Kt = 2.0 for profile keyseat  – Kt = 1.6 for sled keyseat

• Shoulder fillets  – Kt = 2.5 for sharp fillet  – Kt = 1.5 for well-rounded fillet

• Retaining ring grooves  – Kt = Kt = 3.0, or  – Increase diameter by 6%

Forces Acting on A Shaft

Shear and Moment Diagrams From bottom look up

Front view

Shaft Design/Analysis Example SST  : σ 1 − σ 3 ≤

( M  + 32 (  M  π d 

S  y  N 

16

π d 3

2

3

σ  x =

 Mc

τ  xy =

Tc

 I   J 

= =

 M (d  2 )

(π d 

4

64

T (d  2 )

(π d 

4

32

)

)

=

=

32 M 

)

+ T 2 ≤

16 π d 3

( M  −

S  y  N 

⎛ 32 N   ⎞ 3 2 2 d  ≥ ⎜  M  + T  ⎟ ⎜ π S  ⎟ ⎝   y  ⎠

π d 3

16T  π d 3

2

(

2

(

16 ⎛ σ   ⎞ 2 σ 1 = − ⎜  x ⎟ + τ  xy = 3  M  −  M 2 + T 2 2 π d  ⎝  2  ⎠ σ  x

2

1

16 ⎛ σ   ⎞ 2 σ 1 = + ⎜  x ⎟ + τ  xy = 3  M  +  M 2 + T 2 2 π d  ⎝  2  ⎠ σ  x

)

 M  + T  − 2

) )

⎛  ⎜ 32 N  d  ≥ ⎜ ⎜ π  ⎝ 

1

3 ⎛  M  ⎞ ⎛  T   ⎞  ⎞⎟ ⎜ ⎟ +⎜ ⎟ ⎟ ⎜ S  ⎟ ⎜ S  ⎟ ⎟ ⎝   y  ⎠ ⎝   y  ⎠  ⎠ 2

2

Is this correct ?

)

 M  + T  ≤ 2

2

S  y  N 

Fatigue Failure Criterion • Cyclic loading due to shaft rotation  – Find mean and alternating stresses  – Construct Mohr’s circles for mean stress and alternating stress  – Derive effective mean and alternating stresses (based on MSST or DET)  – Use Soderberg or Goodman for design and analysis

Fatigue Failure of Shaft Alternating :

Mean :

σ mx = 0 σ  x = ±

π d 3

τ mxy =

σ  y = 0 τ  xy =

3

π d 

π d 3 σ my = 0

σ my = 0

32 M 

16T 

σ ax =

σ m ' = σ a ' =

16T 

τ mxy = 0

π d 3

32T 

π d 3 32 M 

ANSI/ASME equation for fully reversed bending and steady torsion :

π d 3 Soderberg :

 K t σ a ' S n '

32 M 

+

σ m ' S  y

⎛  σ a  ⎞ ⎜⎜ ⎟⎟ ⎝ S n ' ⎠

=

1

 N 

2

2

⎛  τ m  ⎞ ⎟ =1 +⎜ ⎜ S  ys ⎟ ⎝   ⎠

ANSI Shaft Design Equation

For repeated reversed bending and constant torque

ANSI/ASME Shaft Equation Fully reversed bending and steady torsion : 2

2

⎛  σ a  ⎞ ⎛  τ m  ⎞ ⎟ =1 ⎜⎜ ⎟⎟ + ⎜ ⎜ ⎟ ⎝ S n ' ⎠ ⎝ S  ys  ⎠ Consider safety factor : 2

2

⎛  σ a  ⎞ ⎛  τ m  ⎞ ⎟ =1 ⎜⎜ ⎟⎟ + ⎜ ⎜ ⎟ ⎝ S n ' ⎠ ⎝ S  ys  ⎠ 2

2

2

2

⎛  σ a  ⎞ ⎛  τ m  ⎞ ⎛ σ a N  ⎞ ⎛ τ m N  ⎞ ⎜ ⎟ ⎟ =1 ⎟⎟ + ⎟⎟ + ⎜ ⇒ ⎜⎜ = 1 ⇒ ⎜⎜ ⎜ S   N  ⎟ ⎜ S  ⎟ S   N  S  ' ' n  ys n ⎝   ⎠ ⎝  ⎝   ⎠ ⎝   ys  ⎠  ⎠

ANSI / ASME Shaft Equation Use DET (pure shear for torsion) : 2

2

⎛ σ a N  ⎞ ⎛ τ m N  ⎞ ⎟ ⎜⎜ ⎟⎟ + ⎜ ⎜ ⎟ ⎝  S n '  ⎠ ⎝  S  ys  ⎠

2

⎛   ⎞ 2 2 2 ⎜ ⎟ ⎛ σ a N  ⎞ ⎜ τ m N  ⎟ ⎛ σ a N  ⎞ ⎛  3τ m N  ⎞ ⎟ =1 ⎟⎟ + ⎟⎟ + ⎜ ⇒ ⎜⎜ = 1 ⇒ ⎜⎜ ⎟ ⎜ ⎝  S n '  ⎠ ⎜ 1 S  y ⎟ ⎝  S n '  ⎠ ⎝  S  y  ⎠ ⎜ ⎟ ⎝  3  ⎠

Consider stress concentration : 2

2

2

2

⎛ σ a N  ⎞ ⎛  3τ m N  ⎞ ⎛  K t σ a N  ⎞ ⎛  3τ m N  ⎞ ⎟ =1 ⎜ ⎟ ⎜⎜ ⎟⎟ + ⎟⎟ + ⎜ = 1 ⇒ ⎜⎜ ⎜ ⎟ ⎜ ⎟ ⎝  S n '  ⎠ ⎝  S  y  ⎠ ⎝  S n '  ⎠ ⎝  S  y  ⎠

ANSI / ASME Shaft Equation σ 

a

=

τ m=

 Mc

=

32 M 

 I  π d 3 Tc 16T   J 

=

π d 3 2

⎛  K  σ   N  ⎞ ⇒ ⎜⎜ t  a ⎟⎟ ⎝  S n '  ⎠

2

⎛  3τ m N  ⎞ ⎟ +⎜ ⎜ S  ⎟ ⎝   y  ⎠

2

2

⎛  ⎛ 32 M  ⎞  ⎞ ⎛  ⎛ 16T  ⎞  ⎞  N  ⎟ ⎜ 3 ⎜ 3 ⎟ N  ⎟ ⎜  K t ⎜ 3 ⎟ π  d   ⎠ ⎟ + ⎜ ⎝ π d   ⎠ ⎟ = 1 = 1 ⇒ ⎜ ⎝  ⎜ ⎟ ⎜ ⎟ S  y S n ' ⎜ ⎟ ⎜ ⎟ ⎝   ⎠ ⎝   ⎠

⎡ 32 N  ⎢ d  = ⎢ π  ⎢⎣

1

2

⎛  K t  M  ⎞ 3 ⎛  T   ⎞ ⎜⎜ ⎟⎟ + ⎜ ⎟ ⎜ S  ⎟ S  ' 4 ⎝  n  ⎠ ⎝   y  ⎠

2

⎤3 ⎥ ⎥ ⎥⎦

ANSI / ASME Shaft Equation Fully reversed bending and steady torsion

⎡ 32 N  ⎢ d  ≥ ⎢ π  ⎢⎣

1

2

⎛  K t  M  ⎞ 3 ⎛  T   ⎞ ⎜⎜ ⎟⎟ + ⎜ ⎟ ⎜ S  ⎟ S  ' 4 ⎝  n  ⎠ ⎝   y  ⎠

2

⎤3 ⎥ ⎥ ⎥⎦

Static

⎡ 32 N  ⎢ d  ≥ ⎢ π  ⎢⎣

1

2

⎛  M  ⎞ 3 ⎛  T   ⎞ ⎜ ⎟ + ⎜ ⎟ ⎜ S  ⎟ 4 ⎜ S  ⎟ ⎝   y  ⎠ ⎝   y  ⎠

2

⎤3 ⎥ ⎥ ⎥⎦

ANSI Shaft Design Equation Fully reversed bending and steady torsion :

⎛  σ a  ⎞ ⎜⎜ ⎟⎟ ⎝ S n ' ⎠

2

2

⎛  τ m  ⎞ ⎟ =1 +⎜ ⎜ S  ⎟ ⎝   ys  ⎠

Fully reversed bending and steady torsion

⎡ 32 N  d  ≥ ⎢ ⎢ π  ⎢⎣

Can we use Soderberg or Goodman criterion?

1

⎛  K t  M  ⎞ ⎜⎜ ⎟⎟ ⎝  S n '  ⎠

2

3 ⎛  T   ⎞ + ⎜ ⎟ 4 ⎜⎝ S  y  ⎠⎟

2

⎤3 ⎥ ⎥ ⎥⎦

Example 12-1 (p. 548) • The system transmitting 200 hp from pinion P to gear A, and from pinion C to gear Q. • The shaft rotating speed is 600 rpm. • Shaft material is AISI 1144 OQT 1000

Example (p. 549) • • • •

Free body diagram Shear and moment diagrams Torque at each segment Calculate diameter for locations A, B, C, and D (at both left and right) No moment Torque = 21000

A

No torque

B

C

D

No torque, no moment, vertical shear only

Shear and Moment Diagrams From bottom look up

Front view

Design Examples

Design Example - Torque

Design Example - Forces