Authored by: Urs Kafader Maxon Precision Motor Inc. Key points Resources Maxon Precision Motor Inc., (508) 677-0520, maxonmotorusa.com |
It can be difficult to specify drive components that transform power from a motor into a form that is useful for the load. The overall design may dictate use of basic drive mechanisms such as spindles, gearing, or belts. But the motor or motor/gearing combination determines the parameters that are optimum. These parameters can include spindle pitch, gear reduction ratios, and other factors. Drive-system components, in turn, influence the dynamic response, control quality, and positioning accuracy.
The promulgation of automation has thrust many engineers into the role of specifying drive parameters. So it can be useful to examine the equations that describe how drives transform one form of mechanical power into another: Linear-motion drives convert a motor’s rotary motion into linear motion. Rotary drives convert motor rotation into some other kind of rotation, typically either at a different speed ratio or with an intermittent motion. Linear drives include leadscrews, conveyor belts, cranes, and rack-and-pinion drives. Rotary drives include all types of gearheads and (toothed) belt constructions.
It typically takes just a few parameters to describe the conversion of power in such drives. The more common ones include reduction ratios, efficiency calculations, mass inertia, and a description of mechanical play.
The reduction ratio i indicates the ratio between the angular velocity at the input ω_{in} (or input speed n_{in}) and the angular velocity at the output ω_{out} (or output speed n_{out}). In linear motion drives, we define the reduction ratio as the ratio between ω_{in} (or n_{in}) and the output velocity v_{out}. For calculating input speed the following equations apply:
ω_{in} = ω_{out} • i
or
n_{in} = n_{out} • i π/30 • n_{in} = ω_{in} = v_{out} • i
The reduction ratio for rotary transformation is given as a
dimensionless ratio between two speeds of rotation. (In linear drives the dimension is in units of m^{-1}.)
Next typically comes specification of losses and efficiency. Friction losses in a drive usually get described in a universal fashion by the efficiency, η. The torque required at the drive input is calculated from the required output torque M_{out} or from the output force F_{out} as follows: M_{in} = M_{out}/(i • η), or M_{in} = F_{out}/(i • η)
The usual assumption is that η is constant. This is a rough generalization because efficiency is a function of the load. Alternatively, the friction torque M_{R} can be indicated at the input side as a function of the load and speed. The conversion equations then read:
M_{in} = M_{out}/i + (M_{R}(M_{out}, n_{in})),
or
M_{in} = F_{out}/i + (M_{R} (F_{out}, n_{in} ))
When applications experience acceleration, design calculations must incorporate the masses and moments of mass inertia for the drive elements. Designers must recalculate the moment of inertia with respect to the input axis of the drive, J_{in}. A distinction must also be made between elements that are driven directly by the input shaft (with inertiaJ_{1}), elements located on the output side (with inertia J_{2}), and other moving parts. The mass inertia of these moving parts, J_{3}, must be reflected on the input axis of rotation. The mass inertia so reflected is designated J*_{3}.
On a toothed belt, for example, the appropriate parameters include the mass moments of inertia of the motordriven pulley, J_{1}; of the load pulley, J_{2}; and the mass of the toothed belt, m_{3}, to give:
J_{in} = J_{1} + [(J_{out} + J_{2})/i^{2}] + J *_{3}
A point to note is that the mass inertias of the load side reduce with the square of the drive reduction ratio. This makes the effective mass inertia on the input side much lower. Thus the motor has a much easier time accelerating heavy loads.
Mechanical play, elastic response
The mechanical play, Δφ_{out} , of a drive is conventionally indicated with respect to the output while the input shaft is held fast (blocked input). Backlash indicates the amount of play between drive components. In other words, it is the amount the output can be moved by applying a small amount of force or torque. When there is a change in the direction of rotation, the drive shaft must first compensate for this play, Δφ_{in}, amplified by the reduction ratio before there is any reaction at the output.
For a sufficiently big load at the output, the elastic deformation of the drive elements also come into play and must be added:
Δφ_{in} =Δφ_{out} • i
Thus mechanical play decisively influences the accuracy and quality of the control. It is possible to eliminate mechanical play through use of zero-backlash drives. These apply an elastic preload to drive elements. The downside: This reduction in play comes at the expense of higher losses from friction.
A factor related to mechanical play is elastic play (expressed in terms of rigidity). Elastic play describes how the drive elements elastically deform under load. There can be hysteresis effects accompanying elastic play, depending on the materials used. Most typically, elastic play and hysteresis crop up in drives containing plastic gears, less so in systems exclusively comprised of hard metals. Such effects represent an additional complication. Another point to note is that couplings typically found in drive systems generally exhibit little or no backlash. Their elastic response is fairly rigid with some hysteresis.
When you determine the necessary torque and speed for the input shaft of the drive, your calculations must account for the additional mass inertias, friction, and efficiency of the drive components. The results can be used, in turn, to determine the average thermal load in the form of the rms torque value.
In some cases it only takes a small amount of additional torque to accelerate drive components, compared with the torque needed to drive the load. Here, it is acceptable to make one set of calculations covering steady state operation and during acceleration. Just a few key load parameters are important. They include the maximum speed of rotation, the maximum torque and its duration, and the rms value of the torque (the effective value).
First, consider the calculations for gearheads and toothed belt drives. There are many models and variants of gearheads, from a simple spur gear stage to a Harmonic Drive gear. Standard types are planetary and spur gear heads. The most important selection criteria for these gearheads are output torque, input speed, temperature range, and environmental conditions.
The characteristic variables of a gearhead are its reduction ratio i_{G}, efficiency η_{G}, and moments of inertia J_{1}, J_{2} of the individual gears. Specs for assembled gearheads usually indicate the total inertia with respect to the input shaft, J_{G}. The gear backlash, Δφ_{out}, is given with reference to the output shaft. Here’s how you calculate for:
Speed of rotation: n_{in} = n_{out} • i_{G}
Torque: M_{in} = M_{out}/(i_{G} • η_{G})
Additional torque for constant acceleration (duration Δt_{a} ):
M_{in}, a = (J_{in} + J_{1} + (J_{out} + J_{2})/(i^{2}_{G} • η_{G})) • π/30 • Δn_{in}/Δt_{a}
= (J_{in} + J_{G}+ J_{out}/(i^{2} • η_{G})) • π/30 • Δn_{in}/ Δt_{a}
Play: Δφ_{in} = Δφ_{out} • i_{G}
Maximum efficiency of a spur gear is approximately 90% per stage. This efficiency applies for loads that are typically one-half the rated torque (continuous torque) and greater, where efficiency is essentially independent of the load.
There are other parameters that can be important for specific uses. They include the maximum permissible load of the gearhead, maximum permissible speed of rotation, backlash in positioning applications, temperature range (special lubrication may be needed), whether or not back driving is a possibility in the case of high gear reduction and a large number of stages, and whether or not there can be a reversal in the direction of rotation between input and output.
Toothed belts, compared to gearheads, run more smoothly. They are a frequent choice in uses that demand flexibility in the arrangement of the input and output shafts. Large pulley diameters realize a correspondingly high reduction ratio, a trait that sometimes comes in handy. And it’s easy to configure multiple drives with pulleys. Toothed belts require no lubrication and little maintenance, but are more sensitive to brief impact loads. Backlash can be kept small, although the elastic play is generally higher.
Toothed belts must work in tension, and the pulley bearings must absorb the resulting radial forces. There is also the possibility of generating electrostatic charges if components in the belt drives are not properly grounded. The same can be said for flat belts, V-belts, and chain drives.
Characteristic variables for toothed belts include the diameter of the drive and load pulleys (d_{1}, d_{2}), their moments of inertia (J_{1}, J_{2}), the mass m of the belt , and the total efficiency η. The play Δφout is indicated with reference to the output shaft. Here,
Speed of rotation: n_{in} = n_{out} • d_{1}/d_{2}
Torque: M_{in} = d1/d2 = M_{out}/η
Additional torque for constant acceleration (with duration Δt):
M_{in}, = (J_{in} + J_{1} + (J_{out} + J_{2})/η • (d_{1}/d_{2})^{2} + (m • d^{2}_{1})/(4 • ηG)) • π/30 • Δnin/Δt
Play: Δφ_{in} = Δφ_{out} • d_{1}/d_{2}, plus elastic components
There are friction losses both in the bearings of the pulleys and resulting from deformation of the belt. Sliding friction in the tooth engagement contributes additional losses. But the typical efficiency of toothed belt drives is high, generally running 96 to 98%.
Several parameters can be important when planning toothed belt drives. They include the maximum loading of the belt, which may determine the belt width; a pulley diameter at least large enough to handle the belt’s minimum radius of curvature and the pitch; maximum speed of rotation; and for positioning applications, mechanical play and elasticities.
Sliding spindles, recirculating ball screws
Drive spindles transform rotary motion into linear motion typically by moving a nut up and down a threaded shaft or spindle. The two most common designs are the recirculating ball screw and threaded sliding spindles. Most such spindles have trapezoidal threads, but metric threads are also used in small drives.
Threaded sliding spindles are characterized by high surface pressure, so their nut typically stays in position when there is no power applied. These spindles typically operate at 60% of the duty cycle; the temperature of the nut may become an issue at high rates of operation. Spindles are low-efficiency devices (between 30 and 50%) and generally handle maximum speeds of up to 0.7 m/sec. Materials used in the leadscrew and the nut have an impact on suitability for specific uses. For example, specialty leadscrews made of industrial ceramics can be found in uses characterized by high loads and that demand a long life.
A recirculating ball screw differs from a sliding spindle in that its moving parts ride on balls that roll within guide grooves in the lead and nut. Basically ball screws are threaded spindles with a series of ball bearings that recirculate between the lead and nut. Manufacturers round the profile of the lead to improve the fit with the recirculation balls, so the coefficient of rolling friction is approximately 0.02.
Ball screws are characterized by high repeatability when preloaded, high efficiency (up to 0.99), and correspondingly low heating. Their operating cycles can have duty times to 100%. These devices typically serve in positioning systems and are more expensive than trapezoidal screws. One possible drawback is that ball screws do not self-lock, so a holding brake might be in order for specific uses.
Conversion formulas for screw drives include parameters of thread pitch p, the linear backlash ΔS_{out}, the screw moment of inertia J_{S}, the mass of the nut m_{S}, and the total efficiency η.
Speed of rotation: (in units of min^{-1}): n_{in} = 60/p • V_{out}
Torque: M_{in} = p/2π • F_{out}/η
Additional torque for constant acceleration (over duration Δt_{a}):
M_{in,a} = (J_{in} + J_{S} + (m_{out} + m_{S})/η • p^{2}/4π^{2}) • π/30 • Δn_{in}/Δt_{a}
Play (in radians): Δφ_{in} = ΔS_{out} • 2π/p
The efficiency and friction losses between thread and nut, combined with the friction in the bearings of the screw, result in typical efficiencies between 20 and 50% for a sliding screw, and between 90 to 99% for recirculating ball screws. Applications with play and hysteresis caused by reversing action can minimize these effects by preloading the nut with respect to the screw. However, the value of the preload can influence torque, friction, and operating life of the device.
Other important points in screw drive designs include the maximum loading of the spindle (including operating cycle and heating), maximum speeds of displacement, backlash (for positioning systems), and temperature ranges. Also, screw mounts and bearings in the axial direction must be able to support the transport force over the life of the installation.
As a final example, consider rackand- pinion drives — basically a gear stage where the load pinion has an infinitely large diameter. The radial load of the motor pinion and its bearing system demands special attention because that is where reaction forces arise in the same magnitude as the transport forces.
Variables for the rack-and-pinion drives include the pitch p of the toothing (p = π • T, where T is teeth/in.), the number of teeth z of the pinion, the linear play ΔS_{out} and the mass inertias of the rack m and the pinion J. Total efficiency η is the sum of toothing efficiency (usually 90%) and friction losses in the guide bushings.
Speed of rotation: (in units of min^{-1}):
n_{in} = 60/(p • z) • V_{out}
Torque: M_{in}= (p • z)/2π • F_{out}/η
Additional torque for constant acceleration (over duration Δt_{a}):
M_{in,a} = (J_{in} + J + (m_{out} + m)/η • ((p • z)^{2}/4π^{2})) • π/30 • Δn_{in}/Δt_{a}
Play (in radians): Δφ_{in} = ΔS_{out} • 2π/(p • z)
Important parameters for rackand- pinion drives include the maximum transport force and maximum radial bearing load of the pinion, maximum permissible torques caused by the toothing; maximum speeds, backlash (for positioning systems), lubrication and temperature ranges, and the bearing and guidance of the rack.
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