Leadscrew

Advantages and disadvantages

The advantages of a leadscrew are:

• Large load carrying capability
• Compactness
• Simplicity of design
• Ease of manufacture
• Large mechanical advantage
• Precise and accurate linear motion
• Smooth and quiet operation
• Low maintenance
• Minimal number of parts
• Most are self-locking (cannot be back-driven)

The disadvantages are that most are not very efficient. Due to this low efficiency, they cannot be used in continuous power transmission applications. They also have a high degree of friction on the threads, which can wear the threads out quickly. For square threads, the nut must be replaced; for trapezoidal threads, a split nut may be used to compensate for the wear.

Alternatives

Alternatives to actuation by leadscrew include:

• Ball screws and roller screws (sometimes categorized as types of leadscrew rather than in contradistinction)
• Fluid power (i.e., hydraulics and pneumatics)
• Gear trains (e.g., worm drives, rack-and-pinion drives)
• Electromagnetic actuation (e.g., solenoids)
• Piezoelectric actuation

Mechanics

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 ${\displaystyle \pi d_{\text{m}}}$ 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:

${\displaystyle T_{\text{raise}}=\frac{F{d}_{\text{m}}}{2}\left(\frac{\pi \mu d_{\text{m}}+l}{\pi d_{\text{m}}-\mu l}\right)=\frac{F{d}_{\text{m}}}{2}\tan \left(\varphi +\lambda \right)}$

${\displaystyle T_{\text{lower}}=\frac{F{d}_{\text{m}}}{2}\left(\frac{\pi \mu d_{\text{m}}-l}{\pi d_{\text{m}}+\mu l}\right)=\frac{F{d}_{\text{m}}}{2}\tan \left(\varphi -\lambda \right)}$

where

• ${\displaystyle T}$ = torque
• ${\displaystyle F}$ = load on the screw
• ${\displaystyle d_{\text{m}}}$ = mean diameter
• ${\displaystyle \mu }$ = coefficient of friction (common values are found in the adjacent table)
• ${\displaystyle l}$ = lead
• ${\displaystyle \varphi }$ = angle of friction
• ${\displaystyle \lambda }$ = lead angle

Based on the ${\displaystyle T_{\text{lower}}}$ 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 (${\displaystyle \varphi >\lambda }$). However, this self-locking property is not absolute. Under dynamic conditions—particularly in the presence of vibration or shock loading—the effective coefficient of friction may drop to dynamic levels (which are typically lower than static), allowing the screw to back-drive unexpectedly. For this reason, safety-critical applications often require a secondary holding brake rather than relying solely on the lead screw geometry.

Efficiency

The efficiency, calculated using the torque equations above, is:

${\displaystyle \text{efficiency}=\frac{T_0}{T_{\text{raise}}}=\frac{Fl}{2\pi T_{\text{raise}}}=\frac{\tan \lambda }{\tan \left(\varphi +\lambda \right)}}$

Non-zero thread angle

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 take this into account:

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

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

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

If the leadscrew has a collar 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 (${\displaystyle d_{\text{c}}}$):

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

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

Efficiency for non-zero thread angles can be written as follows:

${\displaystyle \eta =\frac{\cos \alpha -\mu \tan \lambda }{\cos \alpha +\mu \cot \lambda }}$

Running speed

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

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

where

• ${\displaystyle N}$ = critical speed in RPM
• ${\displaystyle d_{\text{r}}}$ = smallest (root) diameter of the leadscrew in inches
• ${\displaystyle L}$ = length between bearing supports in inches
• ${\displaystyle C}$ = .36 for one end fixed, one end free
• ${\displaystyle C}$ = 1.00 for both ends simple
• ${\displaystyle C}$ = 1.47 for one end fixed, one end simple
• ${\displaystyle C}$ = 2.23 for both ends fixed

Alternatively using metric units:

${\displaystyle N=\frac{C{d}_{\text{r}}\times {10}^7}{L^2}}$

where the variables are identical to above, but the values are in millimetres and ${\displaystyle C}$ is as follows:

• ${\displaystyle C}$ = 3.9 for fixed-free supports
• ${\displaystyle C}$ = 12.1 for both ends supported
• ${\displaystyle C}$ = 18.7 for fixed-supported structure
• ${\displaystyle C}$ = 27.2 for both ends fixed