Power screws

The power screws are used to convert the rotary motion into a sliding one (rarely the other way around). They are commonly used as guide screws for machine tools, screws for presses and jacks.


power-screws R 1 R 2 h 3 z H 1 a c a c 30° P Nut thread Screw thread H 4 d 2 d 3 d D 1 D 2 D 4
Fig. 1 - Power screws

Lifting torque:

$M_L=\cfrac{Fd_2}{2}\left[\cfrac{P+\pi\mu d_2sec(30/2)}{\pi d_2-\mu Psec(30/2)}\right]$
where:
$M_L$ lifting torque $\mathrm{Nm}$
$F$ axial force $\mathrm{N}$
$d_2$ medium diameter $\mathrm{mm}$
$P$ thread pitch $\mathrm{mm}$
$\mu$ friction $\mathrm{-}$

Axial stress the screw:

$\sigma=\cfrac{F}{\cfrac{\pi}{4}\left(\cfrac{d_2+d_3}{2}\right)^2}$
where:
$\sigma$ axial stress the screw $\mathrm{MPa}$
$F$ axial force $\mathrm{N}$
$d_2$ medium diameter $\mathrm{mm}$
$d_3$ smaller external thread diameter $\mathrm{mm}$

Shear stress the screw:

$\tau=\cfrac{M}{\cfrac{\pi}{16}\left(\cfrac{d_2+d_3}{2}\right)^3}$
where:
$τ$ shear stress the screw $\mathrm{MPa}$
$M$ torque $\mathrm{Nm}$
$d_2$ medium diameter $\mathrm{mm}$
$d_3$ smaller external thread diameter $\mathrm{mm}$

Maximal shear stress (Tresca) the screw:

$\sigma_{tresca}=\sqrt{\sigma^2+4τ^2}\le\sigma_{Call}$
where:
$\sigma_{tresca}$ maximal shear stress (Tresca) the screw $\mathrm{MPa}$
$\sigma$ axial stress the screw $\mathrm{MPa}$
$τ$ shear stress the screw $\mathrm{MPa}$
$\sigma_{Call}$ allowed combined stress $\mathrm{MPa}$

Allowed combined stress:

$\sigma_{Call}=\cfrac{R_{p0,2/T}}{S_F}\cdot C_c$
where:
$\sigma_{Call}$ allowed combined stress $\mathrm{MPa}$
$R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$
$S_F$ safety factor $\mathrm{MPa}$
$C_c$ coefficient according to load $\mathrm{MPa}$

Coefficient according to load:

load $\mathrm{-}$
Unidirectional load, non-impact load 0,8
Unidirectional load, with a small impact load 0,7
Unidirectional load, with a big impact load 0,6
Alternating load, with a small impact load 0,45
Alternating load, with a big impact load 0,25

Bearing stress the thread:

$p_t=\cfrac{4F}{\cfrac{L_n}{P}\cdot\pi\cdot(d^2-D_1^2)}\le\sigma_{all(t)}$
where:
$p_t$ bearing stress the thread $\mathrm{MPa}$
$F$ axial force $\mathrm{N}$
$L_n$ nut length $\mathrm{mm}$
$P$ thread pitch $\mathrm{mm}$
$d$ thread $\mathrm{mm}$
$D_1$ minor diameter $\mathrm{mm}$
$\sigma_{all(t)}$ allowable bearing stress the thread $\mathrm{MPa}$

Allowable bearing stress the thread:

$\sigma_{all(t)}=\cfrac{0,9R_{p0,2/T}}{S_F}\cdot C_c\cdot C_t$
where:
$\sigma_{all(t)}$ allowable bearing stress the thread $\mathrm{MPa}$
$R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$
$S_F$ safety factor $\mathrm{-}$
$C_c$ coefficient according to load $\mathrm{-}$
$C_t$ power screw coefficient $\mathrm{-}$

Power screw coefficient:

$C_t=2,7083v^2-1,5937v+0,25;\max v=0,25 $
where:
$C_t$ power screw coefficient $\mathrm{-}$
$v$ screw speed $\mathrm{m/s}$

Screw speed:

$v=\cfrac{n}{60}\cdot\pi\cdot d_2$
where:
$v$ screw speed $\mathrm{m/s}$
$n$ speed $\mathrm{rpm}$
$d_2$ medium diameter $\mathrm{mm}$

Buckling:
- Columns with eccentric loading:
The Secant equation for the stress calculation in the extreme fiber of a profile.

$\cfrac{F_{max}}{S}=R_{p0,2/T}/\left[1+\cfrac{ec}{i^2}sec\left(\cfrac{L\cdot\beta}{2i}\sqrt{\cfrac{F_{max}}{ES}}\right)\right]$
applies under the following conditions:
$\cfrac{L\cdot\beta}{i}>0,282\sqrt{\cfrac{ES}{F}}$
$\cfrac{F_{max}}{S_F}\cdot C_c\geq F$
- Struts or short columns with eccentric loading:
$\cfrac{F_{max}}{S}=R_{p0,2/T}/\left[1+\cfrac{ec}{i^2}\right]$
applies under the following conditions:
$\cfrac{L\cdot\beta}{i}\le0,282\sqrt{\cfrac{ES}{F}}$
$\cfrac{F_{max}}{S_F}\cdot C_c\geq F$
where:
$F_{max}$ maximal (critical) force $\mathrm{N}$
$S$ profile area $\mathrm{mm^2}$
$R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$
$e$ eccentricity $\mathrm{mm}$
$c$ extreme fiber distance $\mathrm{mm}$
$i$ gyration radius $\mathrm{mm}$
$L$ strut length $\mathrm{mm}$
$\beta$ type of strut mounting $\mathrm{-}$
$E$ Young’s modulus $\mathrm{MPa}$
$F$ axial force $\mathrm{N}$
$S_F$ safety factor $\mathrm{-}$
$C_c$ coefficient according to load $\mathrm{-}$


buckling F F F F β=0.5 β=0.7 β=1 β=2 A B C D
Fig. 2 - Type of strut mounting

Literature:
- Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas: Konstruování strojních součástí 2010.