New rules for sizing servos

July 12, 2001
Throw out traditional guidelines when matching a servomotor and gearhead.

John Landry
Alpha Gear Drives
Elk Grove Village, Ill.

Edited by Kenneth J. Korane

Servo-style planetary gearheads feature high stiffness, low inertia, high ratios, and high efficiency that permit servomotors to operate at maximum effectiveness.

This typical speed and torque curve for an application is called a trapezoidal speed profile, due to the shape of the speed versus time curve. Inertia is a measure of a body's resistance to acceleration or deceleration. As acceleration time decreases, torque requirements increase. Triangular move profiles can also be used but may be harder for the controller to process. Consult with electrical engineers to determine a best motion profile.

Servomotors provide the ultimate in motion control. The motors precisely control speed and torque, respond in milliseconds, and have higher capacity than traditional dc or ac induction motors of similar size.

Gearboxes routinely mate to servomotors to reduce speed and multiply torque. This lets smaller motors move larger loads and effectively reduce cost, weight, and space. Servo gearheads reduce the effects of inertia as well. That's important because for good system response, motor and load inertias must be similar. This is called inertia matching, or the inertia ratio. If this ratio exceeds 10:1, the servosystem may have difficulty controlling acceleration and deceleration of a load.

Gearboxes offer a significant benefit in that they affect the inertia ratio by a factor of the gearbox ratio squared. This means, for instance, if a load has an arbitrary value of 100 and the gearbox ratio is 5:1, then a motor with an inertia value of 1 would result in an inertia ratio of 4:1. (100 ⁄ 5 2 = 4, or 4:1.) Without the gearbox, the motor would have to be extremely large to accelerate the load at the same rate.

Traditional methods
Traditional gearbox selection methods were developed prior to the advent of servomotors. The process, described by the American Gear Manufacturers Assn. (AGMA), requires an application factor based on the machine, a shock factor related to the type of motor, and a service factor that accounts for the number of cycles per hour. This selection process does not apply to today's servo applications.

The application factor adds capacity for a rock-crusher type gearbox versus a smooth-running conveyor-style gearbox. The shock factor adds capacity related to how smoothly torque transfers from motor to gearbox. The service or safety factor adds capacity to protect the operator should the gearbox fail. None of these factors takes into account the number of expected gearbox cycles. AGMA assumes 1,000,000 cycles equals infinite life, even though systems such as flying cut-offs in printing applications can see 40,000 cycles/hr.

Using the AGMA method in today's servo-driven applications leads to an overall safety factor of 3 to 5. While the gearbox may be suitable, the resulting system is much larger, heavier, and more expensive than necessary.

Selecting a gearbox
Servo gearhead selection should be approached in a more exact fashion. First, determine cycle time — the total time for motion and rest until the motion repeats. This duty cycle can be cyclical or continuous and is important because it determines if gearbox and motor size are based on average or peak torque and speed. Designers should compare this value with the motor's nominal and peak torque and speed ratings. This is easily charted on the torque-versus-speed graphs supplied with all servomotors. If average torque and speed exceed nominal ratings, then the motor requires auxiliary cooling to dissipate generated heat.

Also use torque and speed to determine if the gearbox will dissipate the heat and safely transmit the applied load. If a gearbox supplier does not provide speed ratings in relation to gearhead size, then question the gearhead's capacity in continuous high-speed applications.

And don't assume bigger is better. Larger gears have higher pitch-line velocities and centrifugal forces that limit the effectiveness of the lubrication to adhere to gear teeth. Oil seals also generate more friction and heat at higher speeds and larger shaft diameters.

Duty cycle, D is generally given as a percent,

where tb = acceleration time; tc = run time; td = deceleration time; and te = dwell time. Generally, when duty cycle is less than 60% and each move takes less than 20 minutes, it is considered cyclical. The gearhead can be sized based on the nominal and peak torque and speed.

Movements that exceed 20 minutes may generate more heat than the gearbox can fully dissipate. If not accounted for, this can cause thermal expansion of elements and may seize the bearings or gear teeth. Because servo-planetary gearheads are small compared with traditional gearboxes of equal torque capacity, engineers must analyze thermal expansion of the bearings, housing, shafts, and gears. If parts bolt together or the gears, shafts, and housing have different thermal coefficients of expansion, they may move in relation to each other. As shafts lengthen they exert additional loads on the gearbox or motor bearings.

Also account for metal fatigue when addressing the number of cycles per hour. As the gearbox accelerates and decelerates, gear teeth flex in both directions. If not considered in the gearbox design, the teeth may break. Consult the manufacturers' catalog data to determine the allowable number of peak-acceleration torque and speed cycles in the gearbox ratings.

Rapid reversals in combination with quick accelerations may cause the drive assembly to vibrate if backlash exceeds motor-feedback accuracy. Vibrational loads will greatly reduce the life of the gear teeth and bearings if the gearbox does not have sufficient torsional rigidity to transmit the torque. Higher rigidity and lower backlash simplify control, and gearbox manufacturers should provide rigidity and backlash data.

Torque considerations
After determining cycle time, calculate torque requirements. Designers must determine the system inertia for cyclical applications, with Machinery's Handbook and other publications listing equations for most common shapes.

To maximize reduction in inertia, select the highest ratio gearbox. The rated input speeds at the reducer limit the allowable gearbox ratio and motor size. Multiply the load speed by the ratio to calculate input speed of the gearbox. Also review the required torque at these speeds. Compare calculated speed and torque with catalog values to ensure the motor and gearbox can operate in this range.

Keep in mind that the ratio also has a direct correlation to price. A 10:1 ratio is the largest available in a single reduction in-line planetary gear set. Higher ratios require another gear reduction, which adds cost. The style of the gearhead, in-line or right angle, has an impact as well. Right-angle gearheads are typically more expensive because they are more difficult to assemble, and bevel or face gears cost more.

For in-line planetary gearboxes, the frame size or outside diameter is a limiting feature. This restricts the size of the ring gear, planetary gears, and sun pinion. Helical gears can maximize output, but they increase the load on the planetary bearings in proportion to the helix angle. Size limitations also affect the support bearings. This is most critical in the planetary's small needle bearings that transmit large loads.

Servo gearhead manufacturers generally publish nominal torque, acceleration torque, and possibly an emergency stop (e-stop) torque rating. Size the gearhead by comparing the maximum and nominal torque requirements of a cyclical application to the catalog values. Also calculate the average torque and speed values to take into account the time at various speeds and torques of continuous or highly cyclical applications.

Determine average torque with

where Tm = mean output torque; nb = average output speed during acceleration; Tb = output acceleration torque; nn = average output speed at point "n"; tn = time at point "n"; and Tn = nominal output torque.

Average speed nm comes from

Various manufacturers use different methods to calculate average torque and speed. More-conservative methods calculate the cube root of these values rather than the square root.

Most servocontrolled machines are also have an emergency stop. Be sure to take this torque into account when selecting a gearbox, as an e-stop can be quicker than normal machine decelerations, resulting in large torques due to system inertia. If e-stop torque is not listed in the catalog, be sure to ask the supplier the allowable torque and how many times it can be applied to the gearbox before teeth break.

If a coupling connects the gearbox and load, then a bearing-life calculation is unnecessary. This calculation is only critical if the output shaft drives a belt, universal joint, chain, or gear. Such connections place radial and possibly axial loads on the gear head that can destroy the bearings. The magnitude, location, and duration of the applied load are critical to gearhead life. Look for tapered roller bearings, rather than ball bearings, on the output shaft of the gearhead. They transmit significantly higher radial and axial forces for applications with high output-shaft loads.

This is the basic procedure for selecting a gearhead. However, all manufacturers design their equipment based on different operating principles, so catalog values may vary greatly from supplier to supplier. Analyze the above procedure and question suppliers to compare gearboxes in the same light. The acceleration torque and time applied to the system are critical, and is where most manufacturers ratings differ.

Manufacturers should provide the following data for any servo gearhead. These design factors directly impact a gearbox's performance and life in a given application.

Design life is the number of cycles or hours of life that determines allowable torque and speed values published in the catalog. 10,000 hr, 20,000 hr, 30,000 hr, or 1 million cycles are common ratings. Based on a machine's duty cycle, designers can predict how many years of operation to expect from the gearbox. Ask if the ratings were verified with actual gearbox testing.

Backlash is clearance between gear teeth to compensate for thermal growth and manufacturing errors. Determine if the values are maximum or average. Some manufacturers apply a small load to the gearhead to ensure full-face contact between gears. Ask if the load is a percent of the nominal torque or acceleration torque, what that percentage is, and if the supplier will certify the values.

The relationship between backlash and lost motion is also important. If the manufacturer applies a load to the gearbox to measure backlash, the value can be considered that of lost motion because it takes into account rigidity as well as clearance between the teeth.

Backlash is important in cyclical operations for a couple of reasons. First, it affects the ability of the gearbox to stop at an exact location. System designers use dual-loop feedback and low-backlash gearboxes for such applications. Dual-loop compensation in the controller compares servomotor position with feedback from a critical location on the machine.

Second, too much backlash in a cyclical application may cause gear teeth to prematurely fail. As the teeth accelerate and decelerate, the sides of the teeth in contact constantly change. As clearance between teeth increases, this imparts high impact forces on the gear teeth.

Torsional rigidity, an indication of the stiffness or resistance to twisting in a gearbox, is critical in quick moves. Because servomotors provide millisecond response, any gear teeth deflection or shaft twist affects the controllability of motor speed and position. A gearbox with low rigidity is difficult to control. High rigidity lets the controller accept high inertia ratios, resulting in smaller components and lower cost.

Lubrication: Grease, oil, and synthetic oil all prevent metal-to-metal contact and dissipate heat. Grease is acceptable in low duty cycle applications. Synthetic oil is recommended in highly cyclical applications and required in low backlash operations. Oil dissipates heat the best.

Gearboxes generate heat due to inefficiencies in the bearings, gear teeth, and lubrication, but mainly from friction in the oil seals. While servos and servo gearheads can operate at speeds exceeding 6,000 rpm, gearheads do not have adequate surface area to dissipate frictional heat at these speeds for long periods of time.

Efficiency: Determine if efficiency changes with speed. Low efficiency is inherent in worm gearboxes and some low-backlash designs due to friction and preloading of elements. The motor must provide sufficient torque to overcome internal friction of the gear head. This is also called the no-load running torque, or stiction torque.

Transmission error: Torque or speed ripple (small variations in torque and speed) causes problems in some applications. Ask the manufacturer what values to expect with their products.

Reliability: Review the costs involved in replacing a gearbox, and reference the warranty and design life of a gearhead.

This typical speed and torque curve for an application is called a trapezoidal speed profile, due to the shape of the speed versus time curve. Inertia is a measure of a body's resistance to acceleration or deceleration. As acceleration time decreases, torque requirements increase. Triangular move profiles can also be used but may be harder for the controller to process. Consult with electrical engineers to determine a best motion profile.

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