Rectilinear Vibrating Motion is a back-and-forth linear oscillation produced when a rotating input is converted into straight-line reciprocation along a single axis. The eccentric driver — usually a crank, cam, or unbalanced rotor — sets the stroke length and forces the output member to move forward and back through that fixed displacement on every revolution. It exists to deliver controlled, repeatable shaking along one direction so a tool, hopper, or part can be agitated, fed, or sieved. Industrial vibratory feeders running 1–6 mm strokes at 25–60 Hz move billions of small parts daily on this principle.
Rectilinear Vibrating Motion Interactive Calculator
Vary eccentricity and shaft speed to see stroke, amplitude, vibration frequency, and peak slider speed for a Scotch yoke rectilinear vibrator.
Equation Used
The eccentric crank pin forces the yoke to move in one straight line. The peak-to-peak stroke is twice the eccentricity, so a larger offset creates a longer reciprocating travel. Shaft speed sets vibration frequency, and the ideal sinusoidal peak slider speed is 2*pi*f*e.
- Ideal Scotch yoke with sinusoidal slider displacement.
- Slider motion is constrained to one straight axis.
- Eccentricity is the crank pin offset from the shaft center.
- No allowance for clearance, friction, spring resonance, or structural compliance.
Inside the Rectilinear Vibrating Motion
Rectilinear Vibrating Motion, also called Vibrating Rectilinear Motion in vibration-engineering literature, takes a continuous rotary input and forces a guided slider to oscillate along one straight axis. The classic build is a crank-and-slotted-yoke (Scotch yoke), an eccentric pin running in a guided slot, or a pair of counter-rotating unbalanced masses tuned so their lateral force components cancel and only the inline force remains. Whichever variant you pick, the output stroke equals twice the eccentricity of the crank pin, and the frequency equals the input shaft speed.
The reason this geometry matters is constraint. The output is locked to a linear guide — typically a leaf-spring suspension on industrial feeders, or a pair of linear bearings on machine tools — so the only motion path available is along that one axis. If the guide stiffness drops or a leaf spring cracks, the slider starts to pick up cross-axis wobble and you'll hear it as a buzzy, off-pitch hum instead of a clean tone. On a vibratory feeder, that's the moment parts stop walking forward and start chattering in place.
Tolerances are tight where the crank pin meets the yoke slot. A 0.05 mm clearance is fine; let it open up to 0.2 mm through wear and the slider develops impact noise at each end of stroke, the harmonic content goes ugly, and downstream equipment starts seeing fatigue cracks at the mounts. Common failure modes are spring fatigue from running off-resonance, fastener back-out from running on-resonance without thread-locker, and yoke-slot galling when somebody runs the rig dry of grease.
Key Components
- Eccentric driver (crank or cam): Converts rotation into a linear forcing function. The eccentricity sets stroke — a 3 mm eccentric gives a 6 mm peak-to-peak stroke. Hardened to 58–62 HRC on industrial units to resist pitting at the contact point.
- Yoke or slider follower: Receives the eccentric's force and constrains output to one axis. Slot-to-pin clearance must stay under 0.1 mm — beyond that, end-of-stroke impact noise climbs sharply and bearing life collapses.
- Linear guide or leaf-spring suspension: Holds the slider on its single axis of travel. Leaf springs on feeders are tuned so the natural frequency sits 5–10% below drive frequency, putting the system slightly subresonant for stable amplitude.
- Counterbalance mass (on unbalanced-rotor variants): Cancels reaction force into the frame. Without it, half the input energy ends up shaking the machine base instead of the workpiece, and bolted joints loosen within hours.
- Drive motor: Supplies the rotational input. Sized for peak torque at start-up — typically 3–5× running torque because the spring system has to be pumped up from rest through its resonance band.
Industries That Rely on the Rectilinear Vibrating Motion
Rectilinear Vibrating Motion shows up anywhere a process needs straight-line shaking — feeding parts, sieving aggregate, compacting concrete, or driving a reciprocating cutter. The mechanism is cheap, runs continuously for years, and scales from desktop feeder bowls running a few watts up to mining screens drawing 50 kW. Different industries call the same kinematics by different names; vibratory feeder builders just say "linear feeder", machine-tool engineers describe it through the Scotch yoke, and structural-test labs talk about a "linear shaker".
- Parts feeding: Linear vibratory feeders from Afag and RNA Automation moving fasteners, electronic components, or capsules to pick-and-place stations at 1–6 mm stroke and 50–60 Hz.
- Aggregate processing: Linear motion screens like the Metso TS-series sizing crushed rock at 4–8 mm stroke, 800–1000 RPM, separating fractions for road-base supply.
- Pharmaceutical: Sieve shakers — Retsch AS 200 basic — running rectilinear amplitudes of 0.3–3 mm to fractionate powders into ASTM mesh sizes for QC labs.
- Construction: Concrete vibrating tables and screed beams using twin counter-rotating eccentrics to produce pure inline Vibrating Rectilinear Motion that drives air bubbles out of fresh pours.
- Machine tools: Reciprocating surface grinders and broaching machines using a crank-slider or Scotch yoke to drive the table on cycles up to 60 strokes per minute with 200–800 mm strokes.
- Vibration testing: Electrodynamic and mechanical shakers (LDS V8, Sentek) imposing controlled rectilinear excitation on aerospace components for MIL-STD-810 qualification.
The Formula Behind the Rectilinear Vibrating Motion
The core sizing equation links stroke amplitude and drive frequency to peak slider velocity and acceleration. This is what tells you whether parts will actually walk forward on a feeder, or whether your screen will pass material at the throughput rate the customer ordered. At the low end of the typical operating range — say 20 Hz with a 1 mm stroke — the peak acceleration sits around 1.6 g and parts barely creep. At the high end — 60 Hz with 6 mm stroke — you're producing roughly 43 g, which is plenty for fast feeding but starts beating the daylights out of bearings and spring mounts. The sweet spot for most parts feeders is 4–8 g, which is where you get clean unidirectional part motion without destroying the hardware.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| apeak | Peak acceleration of the slider | m/s² | ft/s² or g (1 g = 32.2 ft/s²) |
| f | Drive frequency | Hz | Hz (cycles per second) |
| S | Stroke (peak-to-peak displacement) | m | in |
| vpeak | Peak slider velocity (derived: vpeak = π × f × S) | m/s | ft/s |
Worked Example: Rectilinear Vibrating Motion in a pharmaceutical capsule linear feeder
You are sizing a linear vibratory feeder to deliver gelatin capsules from a hopper to a blister-pack loader at 300 capsules per minute. The track is 600 mm long, the chosen drive runs at 50 Hz mains, and the leaf-spring stack gives a peak-to-peak stroke of 3 mm at nominal. You need to verify the peak acceleration sits in the 4–8 g feeding window and check the operating-range margins.
Given
- f = 50 Hz
- Snom = 3 mm
- Slow = 1 mm
- Shigh = 6 mm
Solution
Step 1 — convert nominal stroke to metres and find half-stroke amplitude:
Step 2 — compute angular frequency:
Step 3 — peak acceleration at nominal 3 mm stroke:
That's already a touch hot for capsules — they'll feed fast but you'll see bouncing. Now check the low end of the typical range, 1 mm stroke (drive amplitude turned down on the controller):
5 g is right in the middle of the clean-feeding band — capsules walk forward smoothly with no airborne flight phase. Now the high end, 6 mm stroke:
At 30 g the capsules go ballistic — they leave the track surface, tumble, and the orientation work done upstream gets scrambled. Capsules are also fragile enough that 30 g impacts crack the gelatin shell. The actionable answer is: keep the controller setting between 0.8 and 1.5 mm stroke for this build.
Result
Nominal 3 mm stroke at 50 Hz produces 15. 1 g peak acceleration — too aggressive for capsules. Drop the amplitude controller to 1 mm and you land at 5 g, which is the sweet spot for clean unidirectional walking. Pushing to 6 mm gives 30 g, at which point capsules go airborne and shells crack. If you measure feed rate well below predicted at the 1 mm setting, the most likely causes are: (1) leaf-spring natural frequency drifted because a spring cracked or a fastener loosened, putting you off the slightly-subresonant tuning point and killing amplitude, (2) the track angle is below the typical 3–5° downhill bias and friction is canceling the forward component, or (3) capsule loading on the track is over the 30% surface-coverage rule of thumb, where mass damping pulls the resonant amplitude down sharply.
Choosing the Rectilinear Vibrating Motion: Pros and Cons
Rectilinear Vibrating Motion is one of three common ways to move material in a straight line by oscillation. The Vibrating Rectilinear Motion approach competes against rotary-vibratory bowl feeders for parts handling and against pneumatic shakers for heavy material. The right pick depends on stroke, frequency, throughput, and how clean a linear path the application actually needs.
| Property | Rectilinear Vibrating Motion (linear feeder) | Rotary Vibratory Bowl Feeder | Pneumatic Linear Shaker |
|---|---|---|---|
| Typical operating frequency | 25–60 Hz | 50–60 Hz | 5–25 Hz |
| Stroke range | 0.3–8 mm | 0.5–3 mm (helical) | 5–50 mm |
| Peak acceleration window | 4–10 g for parts, up to 50 g for screens | 2–6 g | 1–3 g |
| Throughput rating | 100–1000 parts/min on 100 mm track | 50–600 parts/min, with orientation | Bulk material at tonnes/hour |
| Reliability / lifespan | 20,000+ hrs on leaf springs if tuned correctly | 15,000–20,000 hrs | 8,000–12,000 hrs (pneumatic seal wear) |
| Energy efficiency | High — runs at resonance, 5–50 W typical | Moderate — bowl mass damps gain | Low — compressed air is expensive |
| Cost (entry-level industrial) | $800–3,000 USD | $2,500–8,000 USD (with bowl tooling) | $400–1,200 USD |
| Application fit | Linear delivery to a downstream station | Bulk parts requiring orientation | Heavy material, low duty cycle |
Frequently Asked Questions About Rectilinear Vibrating Motion
You've moved off the resonance peak. A leaf-spring vibratory feeder is tuned to run a few percent below its mechanical natural frequency so the system operates on the rising side of the resonance curve, where amplitude is stable. When you add capsule mass, the natural frequency drops, and now the drive frequency sits above resonance instead of below — gain falls off a cliff.
Fix it by re-tuning the spring stack for the loaded condition, or by adding a closed-loop amplitude controller that adjusts drive voltage to maintain stroke regardless of load. Most modern controllers (Afag SE-series, for example) include this feature.
The formula assumes pure sinusoidal motion at the commanded stroke. Real systems lose amplitude to three things the equation ignores: (1) controller current limiting, where the drive amplifier clips before reaching commanded voltage, (2) spring nonlinearity at large amplitudes — leaf springs stiffen as they bow, reducing actual stroke versus design, and (3) sensor mounting compliance, where a poorly-bonded accelerometer reads low because its base flexes.
Verify by checking the drive current waveform on a scope and inspecting accelerometer mount torque. If both check out, you genuinely are getting less stroke than commanded and the spring stack needs inspection.
For 4 mm at 30 Hz on a continuous-duty machine, counter-rotating unbalanced masses win. They have no sliding contact, so wear-out is bearing-life-limited (40,000+ hours), and you can phase-tune the masses to vary stroke without changing parts.
A Scotch yoke is simpler and cheaper, but the slot-and-pin contact wears measurably within 5,000 hours at this duty, and clearance growth past 0.15 mm puts impact loading into the frame. Eccentric cranks with rod ends are fine for sub-1000 RPM machine-tool reciprocation but overkill for a feeder. Pick on duty cycle: continuous → unbalanced rotor pair; intermittent → Scotch yoke.
You've crossed the throw threshold — peak vertical acceleration exceeded 1 g, so parts go airborne during each cycle. On a flat-track linear feeder, forward conveyance depends on the part staying in contact with the track during the forward stroke and slipping during the return. Once parts launch, they land randomly with no preferred direction, and on a downhill-angled track they actually drift backwards under gravity between hops.
Drop amplitude until peak acceleration sits in the 4–7 g range with the track at 3–5° downhill bias. If you genuinely need higher feed rates, switch to a track angle of 0° and rely purely on micro-throw conveyance, which is stable up to about 9 g.
Thread-locker only works if the bolt preload exceeds the dynamic load amplitude. On a poorly counterbalanced rectilinear vibrator, reaction force into the base can exceed 200 N peak, and cheap M6 fasteners torqued to 8 Nm have preload margins of about that order. You're cycling the bolt joint, and Loctite cures into a brittle film that fractures rather than flows.
Two fixes: (1) upsize fasteners and torque to spec — M8 at 25 Nm gives huge preload margin, and (2) verify the counterbalance phasing on a twin-rotor unit is within 2° of true 180°. A phase error injects unwanted vertical force into the frame that wasn't there in the bench test.
You can — but only if the spring system supports it. A leaf-spring feeder is mechanically tuned to a specific resonance, and varying drive frequency away from that resonance kills amplitude. VFD control works on electromagnetic feeders (where there's no mechanical resonance, just an electromagnet pulling against a return spring at line frequency) and on motor-driven eccentric or unbalanced-rotor systems where the spring suspension is broadly damped.
If you need amplitude control on a tuned spring feeder, vary drive voltage instead — that scales force without moving off resonance. Controllers like the Eriez HiVi do exactly this.
References & Further Reading
- Wikipedia contributors. Vibration. Wikipedia
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