Power Screws

A power screw converts screw or nut rotation into linear displacement. It is used in jacks, presses, clamps, machine-tool feeds, and precision positioners.

Kinematics

Lead and linear speed are

$$s=zp,qquad v=\frac{sn}{60},$$

and the helix angle satisfies \(\tan\gamma=s/(\pi d_2)\).

Torque and efficiency

For an idealized square thread,

$$T_r=F\frac{d_2}{2}\tan(\gamma+\rho),\qquad \rho=\arctan\mu,$$

$$\eta=\frac{\tan\gamma}{\tan(\gamma+\rho)}.$$

Self-locking is approximated by \(\gamma<\rho\), but a safety-critical lifting device still requires a brake or locking element.

Strength and stability

The screw is checked under axial load and torsion:

$$\sigma=\frac{4F}{\pi d_1^2},\quad \tau=\frac{16T}{\pi d_1^3},\quad \sigma_{eq}=\sqrt{\sigma^2+3\tau^2}.$$

A long compression screw also requires Euler buckling and critical-speed checks. Nut threads are checked for bearing pressure, shear, wear, and unequal load distribution. Positioning accuracy further depends on lead error, backlash, elastic deformation, support stiffness, and thermal expansion.

Thread profiles and applications

Square threads offer a simple force model but are difficult to manufacture. Trapezoidal threads are robust and common in motion and lifting screws. Buttress threads suit large predominantly one-directional loads. Ball and roller screws replace sliding with rolling contact for high efficiency and accuracy.

Equivalent friction for trapezoidal threads

For a symmetric thread with half-angle \(\alpha\), flank normal force increases the effective friction. A common approximation is

$$\mu'=\frac{\mu}{\cos\alpha},\qquad \rho'=\arctan\mu'.$$

The lifting-torque equation then uses \(\rho'\) instead of the square-thread friction angle.

Lowering torque and reversibility

For an ideal square thread, the torque required to lower a load is

$$T_{lower}=F\frac{d_2}{2}\tan(\rho-\gamma).$$

A positive value indicates that torque is required to lower the load; if the expression becomes negative, the load can overhaul the screw. Collar or thrust-bearing friction must be added in either direction.

Nut length and load distribution

The first engaged turns carry more load because screw and nut deform differently. Increasing nut length beyond a practical limit provides diminishing benefit. Nut height is selected from bearing pressure, thread shear, wear, manufacturability, and stability of the nut body.

Column stability and critical speed

A compression screw is checked with an effective length determined by its end supports. A rotating slender screw also has a lateral critical speed. Operating speed is limited to a safe fraction of that value and may be constrained further by ball recirculation or permissible \(d_n\) value.

Backlash, preload, and thermal error

Split or double nuts compensate wear and backlash in sliding screws. Ball screws use preloaded nuts to remove axial clearance and increase stiffness. Excess preload raises friction and temperature. Thermal expansion \(\alpha L\Delta T\) can dominate positioning error in long high-duty screws.

Drive power and safety

Required motor power follows from total torque and angular speed:

$$P=T\omega.$$

Include acceleration, seals, guides, and support-bearing friction. Vertical axes need a brake or counterbalance because high-efficiency rolling screws are normally backdrivable.

Design sequence

1. Define force, stroke, speed, duty, accuracy, life, and safety function.
2. Select sliding, ball, or roller screw and thread geometry.
3. Size the screw for combined stress, buckling, and critical speed.
4. Size the nut for pressure, thread strength, wear, and stiffness.
5. Calculate raising and lowering torque, efficiency, power, and heat.
6. Design supports, preload, lubrication, sealing, backlash compensation, and guarding.