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Xiamen Zhisen Electromechanical Equipment Co.,Ltd

ADD: Room 501, No.213 Long Shan South Road, Siming District, Xiamen, China

Contact: Melissa zheng

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Lead screw Torque

The torque required to lift or lower a load can be calculated by "unwrapping" one revolution of a thread. This is most easily described for a square or buttress thread as the thread angle is 0 and has no bearing on the calculations. The unwrapped thread forms a right angle triangle where the base is ${\displaystyle \pi d_{m}}$ long and the height is the lead (pictured to the right). The force of the load is directed downward, the normal force is perpendicular to the hypotenuse of the triangle, the frictional force is directed in the opposite direction of the direction of motion (perpendicular to the normal force or along the hypotenuse), and an imaginary "effort" force is acting horizontally in the direction opposite the direction of the frictional force. Using this free-body diagram the torque required to lift or lower a load can be calculated:[8][9]

${\displaystyle T_{raise}={\frac {Fd_{m}}{2}}\left({\frac {l+\pi \mu d_{m}}{\pi d_{m}-\mu l}}\right)={\frac {Fd_{m}}{2}}\tan {\left(\phi +\lambda \right)}}$
${\displaystyle T_{lower}={\frac {Fd_{m}}{2}}\left({\frac {\pi \mu d_{m}-l}{\pi d_{m}+\mu l}}\right)={\frac {Fd_{m}}{2}}\tan {\left(\phi -\lambda \right)}}$
Coefficient of friction for leadscrew threads[10]
Screw materialNut material
SteelBronzeBrassCast iron
Steel, dry0.15–0.250.15–0.230.15–0.190.15–0.25
Steel, machine oil0.11–0.170.10–0.160.10–0.150.11–0.17
Bronze0.08–0.120.04–0.06-0.06–0.09

where

• T = torque

• F = load on the screw

• dm = mean diameter

• ${\displaystyle \mu \,}$ = coefficient of friction (common values are found in the table to the right)

• l = lead

• ${\displaystyle \phi \,}$ = angle of friction

• ${\displaystyle \lambda \,}$ = lead angle

Based on the Tlower equation it can be found that the screw is self-locking when the coefficient of friction is greater than the tangent of the lead angle. An equivalent comparison is when the friction angle is greater than the lead angle (${\displaystyle \phi >\lambda }$).[11] When this is not true the screw will back-drive, or lower under the weight of the load.[8]

The efficiency, calculated using the torque equations above, is:[12][13]

${\displaystyle {\mbox{efficiency}}={\frac {T_{0}}{T_{raise}}}={\frac {Fl}{2\pi T_{raise}}}={\frac {\tan {\lambda }}{\tan {\left(\phi +\lambda \right)}}}}$

For screws that have a thread angle other than zero, such as a trapezoidal thread, this must be compensated as it increases the frictional forces. The equations below takes this into account:[12][14]

${\displaystyle T_{raise}={\frac {Fd_{m}}{2}}\left({\frac {l+\pi \mu d_{m}\sec {\alpha }}{\pi d_{m}-\mu l\sec {\alpha }}}\right)={\frac {Fd_{m}}{2}}\left({\frac {\mu \sec {\alpha }+\tan {\lambda }}{1-\mu \sec {\alpha }\tan {\lambda }}}\right)}$
${\displaystyle T_{lower}={\frac {Fd_{m}}{2}}\left({\frac {\pi \mu d_{m}\sec {\alpha }-l}{\pi d_{m}+\mu l\sec {\alpha }}}\right)={\frac {Fd_{m}}{2}}\left({\frac {\mu \sec {\alpha }-\tan {\lambda }}{1+\mu \sec {\alpha }\tan {\lambda }}}\right)}$

where ${\displaystyle \alpha \,}$ is one half the thread angle.

If the leadscrew has a collar in which the load rides on then the frictional forces between the interface must be accounted for in the torque calculations as well. For the following equation the load is assumed to be concentrated at the mean collar diameter (dc):[12]

${\displaystyle T_{c}={\frac {F\mu _{c}d_{c}}{2}}}$

where ${\displaystyle \mu _{c}}$ is the coefficient of friction between the collar on the load and dc is the mean collar diameter. For collars that use thrust bearings the frictional loss is negligible and the above equation can be ignored.[15]

Coefficient of friction for thrust collars[15]
Material combinationStarting ${\displaystyle \mu _{c}}$Running ${\displaystyle \mu _{c}}$
Soft steel / cast iron0.170.12
Hardened steel / cast iron0.150.09
Soft steel / bronze0.100.08
Hardened steel / bronze0.080.06

### Running speed

Safe running speeds for various nut materials and loads on a steel screw[16]
Nut materialSafe loads [psi]Speed
Bronze2500–3500Low speed
Bronze1600–250010 fpm
Cast iron1800–25008 fpm
Bronze800–140020–40 fpm
Cast iron600–100020–40 fpm
Bronze150–24050 fpm

The running speed for a leadscrew (or ball screw) is typically limited to, at most, 80% of the calculated critical speed. The critical speed is the speed that excites the natural frequency of the screw. For a steel leadscrew or steel ballscrew, the critical speed is approximately[17]

${\displaystyle N={(4.76\times 10^{6})d_{r}C \over L^{2}}}$

where

• N = critical speed in RPM

• dr = smallest (root) diameter of the leadscrew in inches

• L = length between bearing supports in inches

• C = .36 for one end fixed, one end free

• C = 1.00 for both ends simple

• C = 1.47 for one end fixed, one end simple

• C = 2.23 for both ends fixed

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