Variable Transmission of Motion is any drive arrangement where the output speed, displacement, or velocity profile changes relative to a steady input — either continuously through a variator, in steps through a gearbox, or non-uniformly through a cam or non-circular gear. The Reeves drive on a Bridgeport mill is one of the best-known examples. The purpose is to match a single prime mover to a load whose ideal speed or motion law shifts with operating conditions. The outcome is more torque where you need it, more speed where you don't, and a smoother match between motor and process.
Variable Transmission of Motion Interactive Calculator
Vary the input speed, pitch radii, torque, and efficiency to see the CVT speed ratio, output speed, torque transfer, and belt speed update live.
Equation Used
The calculator uses the article rule that the instantaneous transmission ratio is set by the effective pitch radii. For a belt variator with no slip, belt speed is common to both pulleys, so N_out = N_in x R_driver / R_driven. Torque changes in the opposite direction and is reduced by efficiency eta.
- Belt or variator is not slipping.
- Pitch radii are effective running radii.
- Efficiency is applied to torque transfer.
- Steady-state speed calculation.
The Variable Transmission of Motion in Action
The Variable Transmission of Motion, also called Variable Motion in machine-design textbooks and shop-floor parlance, works by inserting an element between input and output whose effective ratio is not fixed. That element can be a pair of expanding pulleys with a belt riding at a changing pitch radius, a toroidal disc rolling against tilting rollers, a non-circular gear pair where the pitch curve is an ellipse or a higher-order cam profile, or a planetary set whose carrier speed is modulated by a control input. In every case, the same rule applies — the instantaneous ratio is the ratio of effective radii, or for cam-based systems, the derivative of the follower displacement with respect to the input angle.
Why build it this way? Because most loads do not want constant speed. A milling spindle wants 3000 RPM in aluminium and 400 RPM in stainless. A wind turbine wants to keep generator frequency locked while rotor speed wanders with gusts. A textile loom wants the shuttle to dwell, accelerate, dwell again — a non-uniform output speed law that no fixed-ratio drive can produce. Variable Motion lets one motor serve all of those duties.
Tolerances matter. On a Reeves variator, the variable-pitch sheaves must track parallel within 0.2 mm across the belt face — drift past that and the belt walks, the cords fatigue at one edge, and you'll see a belt failure inside 200 hours instead of 4000. On a non-circular gear pair, centre distance must be held to within ±0.05 mm of the design pitch-curve sum, otherwise the mesh tightens at one point in the cycle and backlashes at another. The most common failure modes are belt slip from low clamping force, cam-follower lift-off above the design speed, and shift-actuator stall on stepped variators where the shift fork loses authority under load.
Key Components
- Input shaft and prime mover coupling: Carries steady torque from the motor or engine into the variator. Typical industrial input speeds run 1450-1800 RPM from a 4-pole induction motor; alignment between motor and variator input must hold within 0.1 mm parallel and 0.05°/100 mm angular or shaft seals start weeping inside 500 hours.
- Variable ratio element: The heart of the device — expanding pulleys, toroidal discs, non-circular gears, or a planetary stage with a controlled branch. This element sets the instantaneous ratio. On a Van Doorne push-belt CVT it's a steel band carrying 400+ trapezoidal elements; on a Reeves drive it's a pair of spring-loaded sheaves.
- Ratio control actuator: Moves the variable element to a new ratio. Hand-cranked screw on a manual drive, hydraulic piston on automotive CVTs, servo on modern industrial variators. The actuator must overcome belt clamping force without backdriving — typical hold force on a 7.5 kW Reeves drive is 1.2-2.0 kN per sheave.
- Output shaft and torque sensor (optional): Delivers the modulated speed to the load. On feedback-controlled variators, a torque or speed sensor on the output closes the loop with the ratio actuator so the system holds a target output condition rather than a target ratio.
- Lubrication and traction fluid system: Toroidal CVTs and chain CVTs depend on a traction fluid that solidifies under contact pressure to transmit force without metal-to-metal slip. Fluid film thickness sits around 0.1-0.5 µm; lose film integrity and the contact patch scuffs in seconds.
Who Uses the Variable Transmission of Motion
Variable Transmission of Motion shows up wherever a single power source has to drive a load with shifting demand — speed control on machine tools, ratio control in vehicles, motion-law generation in production machinery, and frequency regulation in renewable energy. The naming changes by industry but the underlying mechanism is the same.
- Machine tools: Reeves drive on a Bridgeport Series 1 vertical mill — variable spindle speed from 60 to 4200 RPM by hand-cranking the variator while the spindle is running. Still in service in thousands of jobshops worldwide.
- Automotive powertrains: Subaru Lineartronic and Toyota K310 push-belt CVTs use a variable speed drive to keep the engine at its best-fuel-consumption point while road speed varies from 0 to 180 km/h.
- Agricultural equipment: Fendt Vario tractors use a hydromechanical Variable Motion drivetrain that splits power between a hydrostatic branch and a mechanical branch, giving 0-60 km/h with no clutching.
- Textile machinery: Picanol air-jet looms use cam-driven non-uniform output speed to dwell the reed during weft insertion then accelerate it for beat-up — a motion law no fixed-ratio drive can produce.
- Wind energy: Voith WinDrive variable-ratio gearbox at the Enercon trial sites used a planetary CVT branch to lock generator output frequency while rotor speed varied with wind.
- Packaging machinery: Bosch sigpack horizontal flow wrappers use elliptical-gear variable transmission to match the cutter knife linear speed to the film web speed during the cut window while reducing it through the rest of the cycle.
The Formula Behind the Variable Transmission of Motion
The instantaneous output speed of any Variable Transmission of Motion is the input speed multiplied by the instantaneous ratio. For a belt-and-sheave variator, the ratio is the pitch-radius ratio; for a non-circular gear pair, it's the ratio of the pitch radii at the current mesh angle. At the low end of the typical operating range the system runs deep into reduction — high torque, low output speed, low efficiency because traction losses scale with contact pressure. At the high end the variator runs near 1:1 or overdrive — low torque, high output speed, peak efficiency. The sweet spot for most industrial variators sits 0.7-1.3 ratio, where contact mechanics, belt life, and efficiency all line up.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| ωout(t) | Instantaneous output angular velocity | rad/s | RPM |
| ωin | Input angular velocity (typically constant) | rad/s | RPM |
| i(t) | Instantaneous transmission ratio | dimensionless | dimensionless |
| rin(t) | Effective input pitch radius at time t | mm | in |
| rout(t) | Effective output pitch radius at time t | mm | in |
Worked Example: Variable Transmission of Motion in a Reeves drive on a glass-bottle inspection conveyor
You are setting the ratio range on a Reeves variable speed drive that links a 4 kW WEG W22 motor running at 1465 RPM to the head pulley of an Emhart Glass FP606 bottle-inspection conveyor at the O-I Glass plant in Alloa, Scotland. The conveyor has to run as slow as 12 m/min for 750 ml wine bottles and as fast as 48 m/min for 200 ml miniatures. Head pulley diameter is 200 mm. You need to confirm the variator's 6:1 range covers it and find the nominal mid-range setting.
Given
- ωin = 1465 RPM
- Dpulley = 200 mm
- vmin = 12 m/min
- vmax = 48 m/min
- Variator range = 6:1 dimensionless
Solution
Step 1 — convert required belt speeds to required head-pulley RPM. Pulley circumference is π × 0.200 = 0.628 m.
Step 2 — at the nominal mid-range condition, target 30 m/min, which is 47.8 RPM at the head pulley. The required variator + fixed-reduction ratio at nominal is:
A Reeves drive alone won't get you 30:1 — you'll need a fixed worm or helical reducer downstream. Pick a SEW R47 helical reducer at 20:1, leaving the variator to cover the residual 1.53:1 nominal with a ±2.5x range either side.
Step 3 — check the low end. At 12 m/min you need total ratio 1465 / 19.1 = 76.7:1, so the variator must run 76.7 / 20 = 3.84:1 reduction. At the high end you need 1465 / 76.4 = 19.2:1 total, so the variator runs 19.2 / 20 = 0.96:1, essentially direct.
The 6:1 variator covers it with margin. At the low-end 3.84:1 setting the belt rides deep on the input sheave and shallow on the output — efficiency drops to about 82% and belt life shortens because side-load on the cords peaks here. At the nominal 1.53:1 the variator hits its efficiency sweet spot near 92%. At the high-end 0.96:1 you're effectively in 1:1 transfer at 96% efficiency, but watch belt-tension fatigue if you sit there permanently.
Result
The variator runs nominally at 1. 53:1 reduction feeding a 20:1 helical reducer, giving 47.8 RPM at the head pulley and 30 m/min belt speed. That's the comfortable middle of the operating range — belt clamping force balanced, no audible whine, full ratio authority either side for product changeovers. At the slow end (12 m/min, 3.84:1 variator setting) the conveyor crawls deliberately and bottle-inspection cameras get full dwell time; at the fast end (48 m/min, 0.96:1) the line feels twitchy and any tracking error on the belt shows up immediately. If you measure 25 m/min when the dial says 30, the three usual suspects are: (1) belt slip from low spring preload on the floating sheave — check that the preload spring still measures within 5% of its free length spec; (2) variator linkage backlash where the indicator dial reads 1.53:1 but the sheaves have crept toward 1.8:1 because the shift screw worked loose; or (3) head-pulley lagging worn down to 195 mm effective diameter, which alone takes 2.5% off belt speed.
Choosing the Variable Transmission of Motion: Pros and Cons
Variable Transmission of Motion competes against fixed-ratio drives with VFD speed control, stepped gearboxes, and direct servo drives. Each option wins on different axes — a VFD on an induction motor is cheaper and simpler if you only need speed change, but a mechanical variator delivers full torque at low speed without the motor heating issues a VFD has below 20 Hz.
| Property | Variable Transmission of Motion (mechanical CVT/variator) | Fixed gearbox + VFD | Stepped gearbox (manual shift) |
|---|---|---|---|
| Output speed range | Continuous, typically 4:1 to 10:1 span | Continuous, 10:1 to 100:1 with VFD limits | Discrete steps, 3-12 ratios |
| Low-speed torque capability | Full motor torque available — no derating | Derated below ~20 Hz unless oversized motor | Full torque in every step |
| Efficiency at nominal | 88-94% (Reeves), 85-91% (toroidal CVT) | 95-97% gearbox × 95-97% VFD ≈ 91-94% | 96-98% |
| Capital cost (per kW, indicative) | £250-£600/kW for industrial variators | £80-£200/kW (motor + VFD + gearbox) | £150-£400/kW depending on number of ratios |
| Maintenance interval | Belt/disc inspection 2000-4000 h, replace 8000-15000 h | Effectively service-free electronics; gearbox oil 20000 h | Gearbox oil 20000 h, shift mechanism check 4000 h |
| Best application fit | Continuous duty with frequent ratio changes under load | Variable-speed pumps, fans, conveyors with light starts | Machine tools with discrete operating points |
| Complexity & failure modes | Belt slip, sheave wear, actuator stall | VFD electronics failure, motor bearing currents | Shift-fork wear, synchroniser failure |
Frequently Asked Questions About Variable Transmission of Motion
That's a classic clamping-force-versus-axial-shift conflict. The shift screw can move the sheave axially, but under high torque the belt's wedging force on the sheaves spikes and locks the moving sheave against the spline. You turn the dial, the screw rotates, but the sheave doesn't translate until you back off the load.
Diagnostic check: measure the dial reading versus actual sheave gap with feeler gauges at no-load and at 80% rated torque. If the gap differs by more than 0.5 mm, the spline is galling or the floating sheave's needle thrust bearing has failed. Replace the bearing — don't just regrease.
Three factors decide it. First, low-speed torque — if the conveyor must start fully loaded or run below 20% of nominal speed for long periods, the variator wins because the induction motor on a VFD overheats at low frequency without forced cooling. Second, ratio change frequency — if you're changing speeds many times per shift under load, the mechanical variator's actuator handles it without the inrush issues VFDs see on rapid speed steps. Third, ambient — VFDs hate dust, condensation, and washdown environments. A sealed Reeves drive in a bottling plant outlasts a VFD cabinet two-to-one in our customer data.
Default to VFD for clean, low-duty-cycle, occasional speed changes. Default to mechanical Variable Motion drive for dirty, heavy-duty, frequently shifted, full-torque-at-low-speed jobs.
Size 30-40% wider than your stated range. Operations always asks for slower than the spec sheet (for new product trials) and faster than the spec sheet (when they're behind on orders). A 4:1 process requirement should be specified as a 6:1 variator. The cost delta between a 4:1 and 6:1 industrial variator is typically 15-20%, far less than retrofitting later.
Don't go past 8:1 in a single mechanical stage though — belt life drops sharply because the deep-reduction setting puts the belt at its minimum bend radius constantly, and cord fatigue shortens replacement intervals from 8000 h to 2000 h.
You're hitting cam-follower lift-off territory. Non-circular gears generate variable angular acceleration through the cycle, and the inertia torque scales with the square of input speed. At 100 RPM the inertia component is small compared to mesh stiffness; at 400 RPM it's 16x higher and the teeth momentarily separate at the acceleration peaks. When they re-engage you hear the rattle.
Two fixes: add output-shaft inertia (a flywheel) to reduce the angular acceleration the gears must impose, or apply a preload through a torsion spring or anti-backlash gear pair. Also check that your centre distance is on spec — a 0.1 mm error on a 100 mm centre distance amplifies the lift-off problem at high speed.
Mechanically yes for belt and toroidal types — the geometry is symmetric. But the control loop usually isn't. Most variators have a ratio-shift actuator sized to overcome the clamping force at rated input torque; if you reverse the power flow, the clamping behaviour and the actuator's load both change sign. You'll find the actuator either backdrives uncontrollably or stalls in the wrong direction.
Push-belt CVTs in particular cannot be reversed without redesigning the hydraulic clamp control. If your application needs bidirectional power flow, specify a hydromechanical split like the Fendt Vario architecture or use a planetary CVT with explicit four-quadrant control.
Yes — they're the same mechanism category. Variable Motion is the older textbook term, used in Reuleaux's nineteenth-century kinematics work and still common in machine-design curricula. Variable Transmission of Motion is the more descriptive modern phrasing favoured in industrial catalogues. Both cover the full family of devices that produce a non-fixed ratio between input and output: belt variators, toroidal CVTs, non-circular gears, planetary split drives, and cam-modulated systems.
Manufacturer quoted figures assume optimum clamping pressure — just enough to prevent slip, no more. In the field, the hydraulic control usually sits 15-25% above the slip threshold as a safety margin, and that extra clamping force translates directly into pumping losses and belt-element friction. You can recover most of the gap by recalibrating the slip-detection algorithm if the controller allows it.
Also check fluid temperature. Push-belt CVTs lose 2-3% efficiency for every 20°C the traction fluid runs above its design point because viscosity drops and the elastohydrodynamic film thins. If your cooler is undersized or fouled, the fluid can sit 30-40°C hot and you'll see exactly the deficit you're describing.
References & Further Reading
- Wikipedia contributors. Continuously variable transmission. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
