# Metrically measured motions

May 5, 2005
The expanding range of metric-sized motion-control components eliminates excuses for not designing to metric standards.

Robert A. Lipsett
Engineering manager
Danaher Motion Inc.
Wood Dale, Ill.

 The availability and variety of motion-control components in metric units of measure is increasing. These metric leadscrews, from Thomson Neff, Danaher Motion for example, are available in 25 sizes with diameters ranging from 10 to 24 mm. It is tough to visually ID metric and English-dimensioned leadscrews, as is evident in this combined photo of both types. Both work the same way and handle the same applications. The only difference is their dimensions. When working with metric, it's important for engineers to keep units straight for calculations of mass, force, and torque. Manufacturer catalogs list most leadscrew efficiencies, but engineers must know what efficiency an application needs. Whether millimeter or inch, forward torque is always positive, while back-driving torque changes. The same calculations work with English units to determine torque. The process of choosing metric components for an application such as a leadscrew is no different than for an English component. A flowchart illustrates the primary factors to know are load, speed, and stroke. This information will help determine the metric leadscrew diameter, lead, nut material, and any support.

The process of designing with metric motion-control components is quite similar to that with their English-dimensioned counterparts. The same design principles and formulas used for English-based measurements apply to metric values. The engineer still needs to know force, lead, and efficiency to calculate torque for a given application. It's just these values are now in metric units of measure.

Torque and force calculations in metric units are often easier and tend to make more sense than their English versions. The metric unit of torque is the Newton-meter (N-m), and the corresponding unit of rotational inertia is kilogram-meter squared (kg-m2.) Leadscrew efficiency is still a function of the helix angle and the friction coefficient. Fortunately, most manufacturers supply efficiency values under differing circumstances in their technical literature.

Many metric leadscrews are new designs so manufacturers may offer special features in their metric line. For example, some metric leadscrews have a modified thread design optimized for plastic nuts. The new threads reduce stress concentrations, especially on high leads.

Factory engineers should note the accuracy used to produce a metric motion-control component. The primary accuracy gage for leadscrews is lead error. Most metric leadscrews offer an accuracy of 300 mm/300 mm with precision levels to 75 mm/300 mm.

Two factors have hampered the adoption of metric component in the U.S. The first is a paucity of components available in metric units. The second is the design engineer's resistance to change. The increasing selection of metric-based products is rapidly overcoming the first obstacle. Engineers must solve the second problem on their own.

Let's walk through a calculation series as an example. Consider a 300 X 300-mm (12 X 12-in.) XY travel table. The table has an upper-axis moving mass of 8.6 kg. The leadscrew has a lead of 5 mm (0.005 m) and a diameter of 18 mm (0.018 m). The manufacturer gives this particular leadscrew an efficiency value of 0.6 when using antibacklash nuts. The table moves a 23-kg payload. This load must accelerate at a rate of 2 m/sec2.

The designer starts by determining the equivalent rotary inertia for the combined mass of the payload and positioning table. The formula to use is:

J = mL2/(e(2p)2)

where J = equivalent rotational inertia in kg-m2, m = total moving mass in kg, L = screw lead in m, and e = screw efficiency.

J = (8.6 + 23) X 0.005 2 / (0.6 X (6.28)2)
J
= .000033 kg-m2, or 33 X 10 6 kg-m 2

The load to be accelerated consists not only of the positioning table and payload, but also includes a manual positioning knob, the motor rotor, a flexible-shaft coupling, and the leadscrew itself. The rotational inertia for each of these items is:

 JKNOB= 6.3 X 10 6 kg-m 2 JCOUPLING= 2.2 X 10 6 kg-m 2 JROTOR= 23 X 10 6 kg-m 2 JSCREW= 36 X 10 6 kg-m2

The designer adds all of these individual values to arrive at the total rotational inertia driven by the motor.

JTOTAL= 100.5 X 10 6 kg-m 2

With total rotational inertia found, the designer calculates torque requirements using the formula:

T = 2pJA/L

where T = the torque in N-m, J = the rotational inertia in kg-m2; A = acceleration in m/sec2; and L = the screw lead in m.

T = 2 X 3.1416 X 100.5 X 10 6 X 2 / 0.005
T
= 0.253 N-m

If the frictional torque of the system is 0.09 N-m, then the total torque required of the motor is 0.34 N-m. The motor chosen for this application has a holding torque of 0.7 N-m that falls off with speed. The intersection of the torque requirements with the motor speed-torque curve determines the load's maximum speed. The motor chosen for this application places this speed at 22 rps or 1,320 rpm.

The original source for this example is from the Motion Control Experts Handbook.

### WHAT MAKES METRIC MEASUREMENTS EASIER?

ENGLISH UNITS
METRIC UNITS
1 yd, 2 ft, 3 1/4 in.
1.607 meters
1 ft, 11 3/16 in.
0.589 m
2 ft, 5 1/2 in.
0.749 m
3 yd, 1 ft, 6 5/8 in.
3.216 m
___ yd, ___ ft, ___ in.
____ m
Add both columns in the table. You should get 6 yd, 2 ft, 29/16 in. for the left column, and 6.161 m on the right. Both distances are the same length. Which one was easier to add?
Source: The U.S. Metric Association

SYMBOL
UNIT NAME
QUANTITY
DEFINITION

m

Meter
Length
base unit
g
Gram
Mass
base unit
s
Second
Time
base unit
K
Kelvin
Temperature
base unit
m2
square meter
Area
m2
m3
cubic meter
Volume
m3
N
Newton
Force
Kg•m/s2
J
Joule
Energy
N•m
W
Watt
Power
J/s

PREFIX

SYMBOL
POWER OF 10
Pico
p
10 12
Nano
n
10 9
Micro
10 6
Milli
m
10 3
Base unit
100
Kilo
k
103
Mega
M
106
Giga
G
109
Tera
T
1012

### What's New in Metric Leadscrews

Many engineers turn to metric leadscrews when designing scanning equipment, waferhandling systems, and data-storage devices, as well as semiconductor and medical equipment. Leadscrews are quiet devices ideal for light to medium loads of less than 650 N. Yet they offer fast and precise movement. The lack of recirculating balls minimizes vibration.

Leadscrews also offer efficiencies that typically range from 30 to 80%, depending on the lead, or helix, angle. Many leadscrews are self-locking at low leads eliminating the need for brakes.

Availability of metric leadscrews is increasing. Engineers can now find them with diameters ranging from 10 to 24 mm and leads from 2 to 45 mm. Accuracy ratings split into standard and precise grades. Standard grades are accurate to 250 m/300 mm. Precision grades deliver 75 mm/300 mm.

### It's a Ten!

Compared to the English system, metric is easy to use. The real problem comes from converting back and forth between the two systems. No wonder, considering the odd mix of ratios in English units of measure. Starting with a simple ounce, it takes 8 oz to equal 1 cup. Two cups define a pint, as do 2 pt a quart. But it takes 4 qt to equal a gallon. As for the barrel measure, it depends on the type of barrel. A barrel of wine is 31.5 gallons, a barrel of beer is 36 gallons, and a barrel of oil is 42 gallons!

In the metric system, there are seven base units of measure or fundamental standards: the meter, kilogram, second, Kelvin, ampere, mole, and candela. The first four are probably the most familiar in the U.S. and represent distance, mass, time, and temperature, respectively. All other metric sizes are derived by multiplying or dividing these base units by a power of 10. There is no need for conversion ratios. For example, the one and only metric unit of length is the meter. Quite a different story from the U.S. units of inches, feet, yards, chains, furlongs, miles, and so forth.

All metric measurements are based on the decimal system. That means starting with a standard base reference, larger and smaller units are expressed as a power of 10. The standard metric measure of volume is the liter, which is slightly larger than a quart. The next size up is 10 liters, called a decaliter. Ten decaliters make a hectoliter and 10 hectoliters create a kiloliter. There are no fractional values or mixed numbers. Every relationship is expressed as a power of 10.

Helping to keep metric numbers from becoming exceedingly large or small was the adoption of a set of metric prefixes. The prefix goes ahead of the base unit and identifies which power of 10 is currently being used as the measurement. The most common prefixes used in engineering step by powers of 1,000.

The U.S. system doesn't stop with only different names. The same name can apply to totally unrelated measurement values. One example is a unit called "the ton." There's a short ton, displacement ton, refrigeration ton, nuclear ton, freight ton, register ton, metric ton, assay ton, and ton of coal, and none are equivalent to each other. According to the U.S. Metric Association, the U.S.'s chaotic collection of confusing units means Americans don't really understand the quantitative information they encounter. Problems especially creep in with mass and force. In the metric system, the unit of mass is the kilogram and the unit of force is the Newton. Weight is the gravitational force on a body and is proportional to its mass: weight = mass X gravity.

A kilogram force is the weight of 1-kg mass and is equal to 9.81 N or 2.2 lb. English units, such as pounds and ounces, are actually units of force despite the fact that kilograms X 2.2 = pounds. The English unit of mass is the slug.

Torque can be another tricky factor. In the metric system, it is expressed as the Newton-meter (N-m). In the English system, it's either the ounce-inch or pound-foot. The metric units for linear and torsional stiffness are Newton/meter (N/m) and Newton/meters/radian (N/m/rad), respectively.

### It's Hard Using Soft Metrics

People in the U.S. are far more familiar with inches and pounds then they are millimeters and kilograms. They don't know the length of 100 mm (about 4 in.) or the weight of 200 g (just under 1/2 lb.) Many find it too taxing and confusing to convert back and forth between systems. Until people build the same awareness with metric that they currently enjoy with English measurements their reluctance to use metric remains.

Federal rules promoting the use of metric measurements haven't helped. Because of the U.S. executive order, U.S. engineers and designers often convert components built in English dimensions to metric measurements without actually doing anything in true metric sizes. Federal contracts require bidders to use metric products measured in rounded units. For example, 300 mm represents 12 in. rather than the exact measurement of 304.8 mm. Some bidding groups state this rule is imposed only to achieve rounded numbers. But, if sent overseas for fabrication the product returns 11.81 in. long. Products built exactly to rounded metric sizes like 300 mm are called hard metric products.

Hard metric production forces manufacturers to retool and may require additional investments in new capital equipment. To minimize these extra costs U.S. manufacturers specify soft metric values. Soft metric values are the exact metric equivalent to the American size. An 8 X 8 X 16-in. concrete masonry block is spec'd as 203 X 203 X 406 mm in soft metric values. The same block in hard metric would be 200 X 200 X 400 mm and thus slightly smaller. U.S. manufacturers use soft metric measurements to "go metric" without changing the current size of their product.

Use of soft metrics is not a proper course of action. Users must know the correct conversion factor for the soft metric value. For example, the SI system has specific values for converting weight (or mass) and force. Engineers may also change the accuracy of a value during the conversion to one the part hadn't originally possessed. Every country except the U.S. works in hard metric values.

### A brief history of METRICS

The Metric system of measurement emerged in 1790 when French scholars set about defining standard units of measure. In fact, guillotine blades used during the French revolution were measured in centimeters, not inches. Common units of measure for commercial needs along with Napoleon and his soldiers helped carry the new system throughout Europe.

In 1789 the first congress of the U.S. debated a national system of weights and measures. Thomas Jefferson, then the first Secretary of State, submitted a proposal for a decimal-based system with a mix of units. His basic unit of measure for distance was the foot, which was slightly less than the traditional foot. Each foot was divided into 10 in. Each inch was divided into 10 lines. And each line into 10 points. Other interesting measurements

in his proposal included the decade, equal to 10 ft; the road at 100 ft; a furlong equaled 1,000 ft; the mile became 10,000 ft. After some debate Congress did what they do best — nothing — leaving the U.S. with a system of carryovers from English weights and measures.

It wasn't until 1866 that Congress legalized the use of the metric system. But businesses did not convert. More than a century later the Metric Conversion Act of 1975 was passed to help jump start voluntary adoption of the metric system by businesses. Few did. In 1988 Congress tried again with the Omnibus Trade and Competitiveness Act. The act mandated all federal agencies specify values in metric units when purchasing materials and equipment. President Bush issued an executive order on the use of metric measurements within the U.S. in 1991. Current U.S. law strongly supports business and industry conversion to metric units; but the ruling is not mandatory in the private sector.

MAKE CONTACT:

Danaher Motion
danahermotion.com
U.S. Metric Assoc.
lamar.colostate.edu/~hillger/

### Robert Repas

Robert serves as Associate Editor - 6 years of service. B.S. Electrical Engineering, Cleveland State University.

Work experience: 18 years teaching electronics, industrial controls, and instrumentation systems at the Nord Advanced Technologies Center, Lorain County Community College. 5 years designing control systems for industrial and agricultural equipment. Primary editor for electrical and motion control.

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