Rotary and longitudinal motion describes the coupled relationship between a rotating input shaft and a straight-line output member, where one type of motion drives the other through a geometric conversion element. The everyday engineering problem this solves is simple — motors spin, but most useful work happens in a straight line. Lead screws, rack and pinion sets, slider cranks, and cam followers each handle that conversion with different speed-to-force ratios. You see it everywhere from CNC ball screws moving cutting heads at 30 m/min to linear actuators lifting 200 lb TV lifts in cabinetry.
Rotary and Longitudinal Motion Interactive Calculator
Vary lead screw pitch and shaft RPM to see the resulting linear travel speed and force tradeoff.
Equation Used
The lead screw converts shaft rotation into linear nut travel. Shaft speed in rpm times screw pitch in mm/rev gives mm/min; dividing by 60 gives mm/s. A larger pitch moves faster at the same rpm, but reduces force multiplication.
- Single-start lead screw with no slip.
- Pitch is axial travel per shaft revolution.
- Efficiency and friction losses are not included.
Operating Principle of the Rotary and Longitudinal Motion
Every rotary-to-longitudinal mechanism does the same job at its core — it takes the angular displacement θ of a driving shaft and produces a linear displacement x of an output member, tied together by a geometric ratio. In a lead screw that ratio is the pitch p, so x = θ × p / (2π). In a rack and pinion it's the pinion radius r, giving x = θ × r. In a slider crank, x is a non-linear function of crank angle and connecting rod length, which is why slider cranks deliver smooth dwell at the dead-centres but never constant velocity. Pick the wrong conversion element for the job and you'll either burn out a motor pushing through bad mechanical advantage, or you'll never hit your positioning accuracy.
The design question always comes back to three things: how much linear force you need at the output, how fast that output must move, and how accurately it must stop. A 5 mm-pitch lead screw spinning at 600 RPM gives you 50 mm/s of linear travel — slow but with serious mechanical advantage. Swap to a 25 mm-pitch ball screw and you hit 250 mm/s at the same shaft speed but you give up roughly 5× of force multiplication. Rack and pinion sits at the high-speed, low-precision end. Cam followers give you exactly the velocity profile you draw on the cam, but only over the cam's stroke length.
When tolerances drift, the symptoms tell you which interface is failing. Backlash in a lead screw shows up as positioning hysteresis — your axis stops 0.05 mm short going one direction and 0.05 mm long going the other. Worn rack teeth produce a clicking pulse once per pinion revolution. Slider crank wrist-pin slop shows up as audible knock at top dead centre. If you measure positioning error worse than 0.1 mm on a system specified to 0.02 mm, the conversion element itself is almost always the culprit, not the motor.
Key Components
- Rotating Input Shaft: The driven shaft turning at angular velocity ω, typically coupled directly to a stepper, servo, or gearmotor output. Runout above 0.02 mm at the coupling face will telegraph straight into the linear output as a once-per-rev wobble.
- Conversion Element: The geometric coupler that ties θ to x — a screw thread, a pinion gear, a cam profile, or a connecting rod. This is the part that defines your speed-force ratio. A 4-start 10 mm-lead screw moves 4× the linear distance per revolution as a 1-start 10 mm-pitch screw.
- Translating Member: The output that moves in a straight line — a nut, a rack, a slider, or a follower. Must be constrained against rotation by a guide rail, anti-rotation key, or linear bearing block. If the constraint flexes more than 0.05 mm under peak load, you lose positioning repeatability.
- Linear Guide: Profiled rail, round shaft, or dovetail that takes side and moment loads off the conversion element. A NSK LH25 rail block holds 30,000 N dynamic load and keeps deflection under 5 µm — without it the screw or rack carries those loads and wears 10× faster.
- End Stops or Limit Sensors: Mechanical or inductive limits that prevent the translating member from over-travelling. Driving a ball nut into a screw end causes ball spillage and total assembly failure — the limit must trigger at least 5 mm before the hard stop.
Real-World Applications of the Rotary and Longitudinal Motion
Rotary-to-longitudinal conversion shows up in nearly every automated machine because rotating motors are cheap and reliable, but the work usually happens in a line. The mechanism choice is driven by the application's accuracy budget and duty cycle — a lead screw for precision, a rack and pinion for speed, a slider crank for cyclic stroke, a cam for shaped velocity profiles. You see all four in the same factory, sometimes on the same machine.
- CNC Machine Tools: Haas VF-2 vertical machining centre uses 32 mm ball screws to position the X and Y axes at up to 1,000 inches per minute with 0.0002 inch positioning repeatability.
- Furniture and AV: FIRGELLI TV Lift Linear Actuators use ACME lead screws to raise a 100 lb flat-screen 1,000 mm out of a cabinet at 12 mm/s with self-locking back-drive resistance.
- Packaging Machinery: Bosch Pack 301 cartoner uses a slider crank to drive the carton-blank push rod through 180 mm stroke at 200 cycles per minute.
- Automotive Manufacturing: Schuler servo presses convert 1,500 RPM rotary motor output into 800 mm vertical ram stroke through a knuckle-joint drive, hitting 30 strokes per minute at 16,000 kN press force.
- Medical Devices: Hill-Rom Centrella hospital bed uses dual lead-screw actuators to raise the deck from 380 mm to 800 mm at 6 mm/s with 250 kg patient load capacity.
- Solar Tracking: Array Technologies DuraTrack HZ v3 uses a rotary slew drive with a longitudinal torque tube to rotate panel rows ±60° to follow the sun across the day.
The Formula Behind the Rotary and Longitudinal Motion
The core relationship for any rotary-to-longitudinal system is the linear velocity v of the output as a function of input shaft speed N and the conversion ratio. For a lead screw or ball screw the conversion ratio is the lead L — the linear distance travelled per full revolution. At the low end of the typical range, say 100 RPM on a 5 mm-lead screw, you get a slow 8.3 mm/s — fine for a TV lift but too slow for a pick-and-place. At the high end, 3,000 RPM on a 25 mm-lead ball screw gives 1,250 mm/s, which is CNC-rapid territory but demands DIN 3 ground accuracy and proper pre-tension. The sweet spot for general automation sits around 500-1,500 RPM with 5-10 mm lead, balancing positioning resolution against thermal growth in the screw shaft.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| v | Linear velocity of the translating member | m/s | in/s |
| N | Rotational speed of the input shaft | RPM | RPM |
| L | Lead — linear distance travelled per full revolution | m/rev | in/rev |
| 60 | Conversion factor from minutes to seconds | s/min | s/min |
Worked Example: Rotary and Longitudinal Motion in a vertical pick-and-place Z-axis on a SMT line
You are sizing the Z-axis lead screw drive on a Juki RX-7 SMT pick-and-place machine that lowers a 0201-component nozzle 25 mm from clear height down to the PCB at 18,000 placements per hour. The motor is a 4,000 RPM servo and you need to choose a screw lead that gives you the cycle time without sacrificing the 5 µm placement repeatability the application demands.
Given
- Nnom = 1500 RPM
- L = 0.005 m/rev
- Stroke = 0.025 m
- Nlow = 750 RPM
- Nhigh = 3000 RPM
Solution
Step 1 — at the nominal operating point of 1,500 RPM on a 5 mm-lead screw, calculate linear velocity:
Step 2 — calculate the time to traverse the 25 mm stroke at nominal speed:
Step 3 — at the low end of the typical operating range (750 RPM, used during fine-placement of fragile 0201 components):
That doubled cycle time is deliberate — at 0.0625 m/s the nozzle settles cleanly without component bounce, which is what you want when placing a 0.4 mm × 0.2 mm chip resistor that costs more in scrap than placement time.
Step 4 — at the high end (3,000 RPM, used for bulk caps and resistors where placement force tolerance is wider):
That's the speed you need to hit 18,000 placements per hour — but at 3,000 RPM on a 5 mm-lead screw you're approaching the screw's DN limit and you'll start seeing thermal growth above 0.01 mm per minute of continuous operation. Above ~3,500 RPM the screw whip becomes audible and positioning repeatability degrades from 5 µm to 15 µm or worse.
Result
Nominal Z-axis traverse time is 0. 20 s for the 25 mm stroke at 1,500 RPM. In practice this feels like a quick down-stroke that a human eye can just barely track — fast enough for 18,000 placements/hour with margin for vision-system settling. The low-end 750 RPM operation doubles cycle time to 0.40 s but buys you the gentle placement profile fragile parts need, while the 3,000 RPM high-end gets you down to 0.10 s but pushes the screw into thermal-growth and whip territory. If you measure traverse time longer than 0.20 s at nominal speed, the most common causes are: (1) servo torque saturation because the ball nut preload is set above 8% of dynamic load rating, (2) coupling slip at the motor-screw interface from an under-torqued clamp collar, or (3) screw shaft bow from improper end-bearing alignment greater than 0.05 mm TIR.
Choosing the Rotary and Longitudinal Motion: Pros and Cons
Choosing between rotary-to-longitudinal mechanisms is rarely about which is best — it's about which fits your speed, force, accuracy, and budget envelope. Here's how the four common options stack up on the engineering dimensions that actually drive the decision.
| Property | Lead Screw / Ball Screw | Rack and Pinion | Slider Crank |
|---|---|---|---|
| Typical linear speed | 50-1,500 mm/s | Up to 5,000 mm/s | Stroke-limited, up to 1,000 cycles/min |
| Positioning accuracy | 5-50 µm | 50-200 µm | Not a positioning device — fixed stroke |
| Load capacity | Up to 50,000 N axial | Up to 25,000 N continuous | Defined by crank size, up to 1,500 kN in presses |
| Cost (relative) | Medium to high | Medium | Low |
| Maintenance interval | Re-lube every 10,000 km of travel | Inspect teeth every 2,000 hours | Wrist pin and connecting rod every 20,000 hours |
| Best application fit | Precision positioning, vertical loads | Long-travel high-speed gantries | High-cycle reciprocating motion |
| Backlash | 0-5 µm with preloaded ball nut | 20-100 µm depending on tooth quality | None — kinematically constrained |
Frequently Asked Questions About Rotary and Longitudinal Motion
That's classic backlash — the gap between the screw thread flanks and the nut thread flanks. When you reverse direction, the motor has to spin through that gap before the nut starts moving again. On a standard ACME nut you'll see 50-150 µm of backlash. A preloaded ball nut drops it to under 5 µm.
Quick diagnostic: command the axis to position 100 mm from the negative side, mark it, then command 100 mm from the positive side. The offset between marks is your bidirectional backlash. If it grows over time, the nut is wearing — replace before it exceeds your accuracy spec.
You're hitting the screw's critical speed, also called whip speed. The screw shaft is essentially a slender beam, and above its first natural frequency it starts to whip in a sinusoidal bow between the end bearings. The translating nut rides that bow and your positioning becomes a function of where in the whip cycle you stop.
Critical speed scales with d / L2, so longer screws hit it sooner. Fixes are: (1) shorten the unsupported length with an intermediate bearing, (2) go to a larger shaft diameter, or (3) reduce max RPM and use a higher-lead screw to keep linear speed up.
At 3 metres, rack and pinion is almost always the right answer. Ball screws scale poorly with length — critical speed drops as 1/L2, and a 3 m ball screw would limit you to maybe 1,500 RPM with intermediate supports, plus the screw cost goes up dramatically with grade and length.
A precision ground rack section bolts up modularly — you can run 30 metres of travel by butting rack sections, and your linear speed is limited only by pinion RPM and motor torque. The trade is positioning accuracy: expect 50-100 µm with a quality helical rack versus 10-20 µm with a ground ball screw. If your application is laser cutting or plasma, that's fine. If it's optical metrology, stick with the screw and accept the length penalty.
The textbook stroke = 2R formula assumes the connecting rod is infinitely long. Real connecting rods have finite length, and the geometry shortens slightly at top dead centre versus bottom dead centre. The deviation is small — typically under 0.5% for a rod-to-crank ratio of 4:1 or higher — so 2 mm short on a stroke of 200 mm sounds like something else.
Check wrist-pin clearance and big-end bearing clearance first. Combined slop of 1 mm at each end is enough to lose 2 mm of effective stroke. Also verify your TDC and BDC are being measured at the actual extremes, not at electrical limit-switch trip points which usually sit a few mm inside the mechanical extremes.
Cam-follower wins when the velocity profile is fixed, the cycle rate is high (above 200 cycles/min), and the duty cycle is essentially 100%. A well-cut cam reproduces its profile to within 10 µm forever, with no controller, no tuning, and no software to debug. Bosch packaging machines and rotary indexers from companies like CAMCO use cams for exactly this reason.
Servo + lead screw wins when you need to change the profile in software, when stroke length must vary part-to-part, or when the cycle rate is below ~100/min where servo settling time is not the bottleneck. Rule of thumb: if you'd run the same motion for 5 years without changing it, cut a cam. If you'd change it next month, use a servo.
Almost certainly the dynamic torque demand during acceleration. Static load tells you the torque to hold position; it doesn't tell you the torque to accelerate the load mass plus the screw's own rotational inertia from zero to your commanded speed.
Add the inertial torque: Taccel = Jtotal × α, where Jtotal includes both the reflected linear mass (m × (L / 2π)2) and the screw shaft inertia. On long screws the shaft inertia alone can dominate — a 2 m × 32 mm screw has roughly 0.008 kg·m2, which at 100 rad/s2 needs 0.8 Nm just to spin the screw, before any payload. Re-run your sizing with peak torque, not RMS, and you'll usually find the next motor frame size up solves it.
References & Further Reading
- Wikipedia contributors. Linear motion. Wikipedia
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