mechanical systems design topic readings

Mechanical Systems DesignTopic Readings

Steven H. Collins

February 18, 2021

Chapter 12

Catalog Component Selection

Some assembly functions are best accomplished using components purchasedfrom a catalog. Such components typically perform one specific function com-mon to many machines, usually a function that requires relatively high complexityand precision for adequate performance. Custom designs of such elements wouldbe expensive and time consuming. Catalog components provide efficient solu-tions through the economies of mass-manufacture, with the trade-off that only adiscrete set of predetermined designs are available. In other words, catalog partsperform the most difficult, but common, tasks, while custom parts hold everythingtogether nicely for a particular application.

Designing with catalog components requires selection across a discrete set. Firstwe need to know the types of things that are available and the properties that willbe important for each. We can then use analysis of simplified models of the systemto determine good values for each property. Catalog components are not alwaysavailable with the desired properties, sometimes due to Physics and other timesdue to lack of customer interest, so we’ll have to iterate.

In this chapter, we will provide guidelines for the selection and integration of afew catalog components common to robotics: fasteners, bearings, shafts and bat-teries. We will also touch lightly on gears and motors, each of which is covered indetail in another chapter. This is not intended as a replacement for a detailed anal-ysis of machine components such as found in, e.g., Budynas and Nisbett [2006]chapters 7-14. Instead, this serves as a practical guide for those getting startedwith mechanical systems design, with an expectation that you will later consultother sources for more detailed models and empirical data as necessary.

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12.1. FASTENERS 3

12.1 FastenersFasteners are used to create rigid, non-permanent joints between parts. Here wewill consider two common types: socket cap screws and bolts (with nuts).

12.1.1 Machine ScrewsHere are some important terms used to describe screws:

• Threads are the helical grooves on the outside of the screw (external threads)that allow it to interlock with the matching grooves inside a nut or threadedhole (internal threads). Most screws are single threaded, having one contin-uous groove rather than several parallel grooves, and right-handed, meaningthat clockwise rotation will cause the screw to move into the hole.

• Pitch indicates the distance between adjacent threads, commonly given asthe inverse of this value, e.g. threads per inch.

• Major Diameter is the diameter of the outer edge of the threads, equivalentto the diameter of a hole through which the screw would just barely fit.

• Tap (or Die) Diameter is the diameter of the hole (or positive cylindricalfeature) that should be manufactured to allow internal threads (or externalthreads) to later be cut into the material using a tap (or die).

An 8-32 stainless steel socket cap screwwith a hex drive and partial threads.

Standards for thread shape and tolerance are set by the Unified Thread Standard,or UTS, for inch screws, or the International Organization for Standardization,or ISO, for metric screws. Two common inch thread designations are UNC, orcoarse, with fewer, stronger threads per inch, UNF, or fine, with more threads andprecision (UNF).

4 CHAPTER 12. CATALOG COMPONENT SELECTION

Threads are usually specified as D-P, where D relates to major diameter and Pis thread pitch. For example, a 6-32 screw has an outer diameter ‘number’ or‘gage’ of 6 and 32 threads per inch. Larger gage screws are bigger. Thread typesand corresponding major diameters, clearance hole diameters, and tap diameterscan be found in a Drill and Tap Chart.

Example of a tap and drill chart.

Screw strength will depend on its size and material. Peak rated load is an estimateof the axial tension at which a screw will break. Failure load ratings are listed bysuppliers or in UTS specifications. For example, the rated failure loads of Grade 2,hardened steel socket cap screws are about:

4-40 200 lbf6-32 300 lbf8-32 450 lbf

10-24 600 lbf1/4-20 1000 lbf

12.1. FASTENERS 5

As a screw is tightened, it becomes loaded in tension. Torque applied to the headof the screw tends to cause rotation of the threads, which tend to pull the threadedregion deeper into the hole, stretching the screw and loading it axially. To relateapplied torque to axial tension, we can model the threads of the screw like a wedge(see the figure below). Torque tends to drive this wedge under the accepting part,pulling it towards the screw cap. The driving force on the wedge is approximatedas the torque divided by the thread radius, resisted by normal and frictional forceat the thread interface. The axial components of these normal and frictional forcesare counteracted by an axial force between the screw and accepting part:

Faxial ≈τ

1/2 ·dm· cosθ −µ sinθ

µ cosθ + sinθ(12.1)

where τ is the torque applied to the screw, dm is the major diameter, µ is thecoefficient of friction. The angle of the wedge is

θ = arctan(

lp

π ·dm

)(12.2)

where lp is the pitch length. For typical values of µ and θ , say 0.3 and 0.07radians, the second term evaluates to about 2.5, or

Faxial ≈ 5 · τ

dm(12.3)

If the screw is at rest, friction will instead act to prevent the screw from spinningout. During loosening, friction will resist the applied torque, acting in the oppositedirection. In the special case that friction is negligible, such as with a ball screwor well-lubricated power screw, this relationship reduces to:

Faxial ≈ 2π · τ

lp(12.4)

Screw torque and axial loading related by modeling threads as a wedge.


Axial tension can also be estimated by calculating the stretch of the screw, equalto the number of rotations times the pitch length, and its stiffness which is a func-tion of elastic modulus, E, cross-sectional area, A, and resting length, L.

Axial loads induced by tightening a screw are usually helpful. Screw forces causea reaction force between the fastened and threaded parts. Friction at the contactwill prevent the fastened part from becoming misaligned. When the fastened partis loaded, the screw will typically not experience additional loading; instead, thereaction force between the fastened part and the threaded part will decrease. It isusually best to tighten a screw until the induced axial load is moderately higherthan the expected axial load transferred through the assembly.

A common failure mode for screws and threaded holes is failure of the threaditself, also known as stripping of the threads. A simple heuristic to avoid strippingthreads is to always have at least three full threads in contact; most load transferoccurs in these first three threads regardless of the total thread number. In mostsituations, with three or more full threads the screw itself will fail in tension be-fore the threads are stripped. Additional threads can be added if the material ofthe threaded part is significantly weaker than that of the screw. If it is importantfor a screw to remain perpendicular to the surface into which it is inserted, a goodheuristic is to provide threads to a depth of at least two screw diameters.

12.1.2 Nuts and Bolts

In most robotics applications, it is preferable to use a machine screw with athrough hole and a threaded hole in a custom part rather than a bolt with a nut andtwo through holes. Threaded holes result in fewer parts and lower mass. Threadedholes also result in lower expected error, since they maintain screw position moreprecisely than a bolt in a clearance hole. In some cases, however, threading holesmay be prohibitively expensive, and nuts and bolts are then a useful alternative.

Nuts and bolts are designed to be tightened strongly and precisely during instal-lation. In a typical application, the nut and bolt drive two assembled elementstogether with enough force that friction between the two transfers all expectedloads. During the tightening process, the nut will often yield slightly, and it istherefore best not to reuse nuts once fully tightened. Bolts nearly universallycome with hex heads, and nuts typically have an external hex shape with aboutthree full internal threads. Concepts related to threads, standards, tensioning, andstrength are otherwise very similar for bolts and machine screws.

12.2. BEARINGS 7

12.1.3 Fastener Selection Process

To select a machine screw or nut and bolt:

1. Determine the required axial strength using free-body-diagram analysis ofthe mating components. Be sure to account for coefficient of friction, if theload is transferred through contact friction, and for factor of safety.

2. Consult a screw strength chart to determine an appropriate major diameter(gage) and material. If the screw and threaded hole materials are dissimilar,strength will be set by the weaker material.

3. Consult a catalog, such as McMaster-Carr or Bolt Depot, to find a part withthe desired diameter and material, a common thread pitch, and other prop-erties appropriate to your application such as length, head and drive type.

4. Check that the rated strength, and other properties, of the final part meetyour requirements and iterate as necessary.

12.2 Bearings

Bearings connect parts together while allowing relative motion. Here we willdiscuss rotational, thrust, linear and specialty bearings.

12.2.1 Rotational Bearings

Single degree of freedom rotational bearings, used in forming ‘revolute’ (or ‘pin’or ‘hinge’) joints, are the most common and well-developed type of bearing. Theytend to result in simple, lightweight, high-performance designs. It is advisable tofirst try to generate desired assembly motions using revolute joints.

Left to right: Plain bearing (or bushing), ball bearings, and needle roller bearing.


The most important properties to consider when selecting rotational bearings arestrength, size, resistance, speed, and precision, which are typically reported usingthe following terms:

• Radial load is the maximum allowable force applied to the bearing in thedirection orthogonal to the axis of rotation.

• Axial load is the maximum allowable force applied to the bearing in thedirection along the axis of rotation. It is usually much lower than the allow-able radial load.

• Inner diameter, or shaft diameter, is the diameter of the hole in the bearing(or its inner race) into which a shaft is inserted.

• Outer diameter is the diameter of the outer surface of the bearing.

• Width, or axial length, is the length of the bearing along the axis of rotation.

• Coefficient of friction is the effective coefficient of friction for calculatingturning resistance.

• Rated speed is the maximum rotational speed allowable due to the dynam-ics of internal rolling elements in ball bearings.

• Precision of a bearing relates to the expected error in shaft position due toeccentricity, gaps or compliance. Ball bearing tolerances are standardizedto the ABEC scale, with higher ratings indicating better precision.

Resistance to rotation in a bearing can be modeled as Coulomb friction:

τ f ≈12·Di ·µ ·Fradial (12.5)

where τ f is the resistance torque due to friction, Di is the inner diameter, µ is theeffective coefficient of friction, and Fradial is the radial load. The friction torquewill act to oppose movement. At high speeds, viscous damping can also occur,modeled as τv ≈−CD · θ̇ 2, where θ̇ is shaft velocity.

The most common forms of single degree of freedom rotational bearings are plainbearings, ball bearings, and needle roller bearings:

• Plain bearings, or bushings, are often no more than a thin tube made froma material with low friction and good abrasion resistance. Plain bearingsare small, lightweight and cheap, and typically tolerate high loads. Theyare less precise than ball bearings, however, due to the need for clearancebetween the hole and shaft. They also tend to have higher resistance, sincerelative motion is achieved through sliding rather than rolling.

12.2. BEARINGS 9

Maximum radial load is usually given as a peak pressure in plain bear-ings, which can be related to peak radial force using the approximationσ ≈ Fradial · (w ·Di)

−1, where w is width and Di is inner diameter. Effectivevalues of µ range from about 0.05 for Teflon to 0.20 for lubricated brass.

Plain bearings should be of a dissimilar material to the shaft to prevent ad-hesion. Common bushing materials include metals, like brass, and plastics,like Teflon. Brass is cheap and durable, but has high friction and density.Teflon has low friction, but also low strength and abrasion resistance, and istherefore often combined with brass or another strong backing element.

• Ball bearings comprise a set of rolling spherical elements housed betweenan inner and outer ‘race’. They have very low friction (µ ≈ 0.001), highprecision and tolerate high speeds. However, contact stresses on the rollingelements lead to large, heavy parts for a given radial load. For example,with radial loads on the order of 100 pounds force, ball bearings are typi-cally about 10 times heavier than a bushing.

Ball bearings will quickly break if axial loads are applied to the inner race.The balls roll in a shallow groove which affords little positive traction againstaxial motion and acts as a force multiplier from axial to radial load. Simi-larly, small off-axis moments on the inner race will damage a ball bearing.

Shielding creates a barrier between the environment and the moving ele-ments inside a ball bearing. This prevents damage from particles that couldenter the bearing and lodge between the ball and the race, but also increasesresistance, width and mass.

• Needle roller bearings are similar to ball bearings, with cylindrical ratherthan spherical rolling elements. Increased contact area between the rollersand the race allows higher radial loads in a smaller, lighter component.These cylinders are harder to assemble with an inner race, however, andsmall-diameter roller bearings typically do not have one. The mating shaftshould therefore be prepared for direct contact with the cylinders, prefer-ably comprising hardened, polished steel. Precision is thereby poorer insmall diameter roller bearings due to the need for clearance between therollers and the shaft during assembly.

The choice among plain, ball and roller bearings will depend on the relative im-portance of resistance, size, mass, precision and cost. The implications may notbe obvious, however, and so it is usually a good idea to spec out one of each andcompare their properties.


Radial bearings can be seated in a part using retaining compound or press fit.If gluing, the hole should have a slip fit (slightly oversized). Press fitting involvespressing the bearing into a hole that is slightly undersized. It is difficult to getright and requires precise fabrication (e.g. reaming) of the hole.

Flanges are a common option in radial bearings, and can help seat the bearingwith precise axial position. In plain bearings, the flange can have the added bene-fit of acting as a built-in thrust washer.

12.2.2 Thrust BearingsThrust bearings, or thrust washers, are often used in combination with radial bear-ings to fully constrain the elements of a revolute joint. Thrust bearings take axialloads that would damage radial bearings. They are usually placed such that oneface contacts a custom part housing the bearing, while the other face contacts anelement attached to the shaft, such as a gear or a second custom part. Thrust bear-ings are available with designs analogous to their rotational counterparts, includ-ing plain, ball and roller thrust bearings, with corresponding qualities. Rollingthrust bearings require an appropriate rolling surface, which can be built into acustom part or provided with separate washer elements purchased with the bear-ing. Whenever significant axial loads are expected, some explicit form of thrustbearing is advisable to manage friction and wear.

Left to right: Thrust washer, roller thrust bearings, and needle thrust bearing.

12.2.3 Linear BearingsLinear bearings and rails are used to create translational joints between compo-nents. The primary sliding and rolling elements are analogous to their rotationalcounterparts, including plain and ball bearings. Joints designed around linearbearings have numerous disadvantages compared to their rotational counterparts,however, and tend to result in much larger, heavier, more complex, and more ex-pensive designs. Drawbacks include:

12.2. BEARINGS 11

Linear bearings and rails. Top left: linear plain bearings. Top right: roundlinear bearing with recirculating balls. Bottom left: round linear bearings on rails.Bottom right: rectangular linear bearing on a multi-featured rotation-resisting rail.

• Constraining a joint to have only one linear degree of freedom is difficult.Bearings on round rails will rotate in addition to translating, necessitatingmultiple rails. Rails with features that resist rotation are complex and large.To prevent binding, linear bearings must usually be relatively long. Largeoff-axis moments often occur, for example from offset driving elements,requiring multiple linear bearings spaced out along each rail.

• Recirculating rolling elements are required in linear ball or roller bear-ings, because each rolling element travels at half the speed of the bearingas it translates along the rail. This requires separate channels, resulting in alarger and more complex housing than its rotational counterpart.

• Large, precise rails add to the size, mass and cost of linear joints. Theentire rail surface must be hard and smooth, since rolling or sliding elementscontact the rails directly. Rails must have a large cross-section to keep themstiff and avoid deflection and binding. Rails must be long enough to allowadequate travel, accounting for bearing length, and most of the length isunused most of the time.

To summarize, a well-designed linear joint will typically require four large, com-plex linear bearings with a large square footprint, riding on two long, thick, pol-ished rails, making the joint large, heavy, complex and expensive.


12.2.4 Specialty and Combination BearingsIn some applications, other types of bearings could provide an advantage:

• Angular contact bearings are a hybrid of radial and thrust bearings, inwhich the moving elements roll on a conical surface. They are designedfor situations in which a consistent combination of radial and axial load areexpected, such as might arise in a set of bevel gears.

• U-joints, or gimbals, allow two rotational degrees of freedom. They arecomprised of two revolute joints with orthogonal, intersecting axes of rota-tion, typically achieved via a + shaped ’floating’ element in simple supportby two bearing sets, one on each component being joined.

• Ball joints, or ball and socket joints, allow three rotational degrees of free-dom. They are comprised of a spherical element partially captured by aspherical enclosure. Ball joints are usually used in concert with additionalconstraining elements.

• Gantries allow two translational degrees of freedom. They are usually cre-ated by mounting the ends of the rails of one linear joint onto the bearingsof a linear joint with an orthogonal axis. They multiply, rather than add, theills of linear joints. Nevertheless, sometimes they are worthwhile.

12.2.5 Bearing Selection ProcessTo select a bearing:

1. Use free-body-diagrams of the shaft and connecting elements (e.g. gears)to determine the maximum radial and axial loads.

2. Use the above concepts to guess which type of bearing is suitable.

3. Consult a catalog, such as McMaster-Carr, for candidate parts, using bear-ing type and radial load to constrain the search.

4. Select among the available options based on secondary considerations suchas flanges or shielding.

5. Note outcomes such as size, mass, precision and cost and calculate the ex-pected resistance torque.

6. Go through steps 3–5 again for a different type of bearing (plain, ball orroller) and compare results. Iterate as necessary.

12.3. SHAFTS 13

12.3 ShaftsShafts are cylindrical components used in combination with bearings to allow ro-tational motion, and often used in combination with elements like gears to transmittorques. Since the desired shaft length and features are unique for every appli-cation, shafting, or shaft stock material, is usually purchased and cut to length,rather than purchasing fully featured components. The most important propertiesin shaft selection are material and outer diameter.

Shafts are often loaded in torsion and/or bending. Rearranging terms from a sim-plified analysis of peak stress in these cases and solving for diameter, we have:

D ≈(

16 ·T · fosπ ·Sys

)1/3

, and D ≈(

32 ·M · fosπ ·Syt

)1/3

, (12.6)

where D is the shaft diameter, T is the torque, fos is factor of safety, Sys is shearyield strength, and Syt is the tensile yield strength. For steel, Sys ≈ 1/2 ·Syt . For agiven material and loading, you can thereby quickly select the shaft diameter.

In applications where rolling elements make direct contact with the shaft, suchas needle roller bearings or linear rolling bearings, it may be necessary for theshaft surface to be hard and precise. Shafting for these applications is typicallymade from hardened, ground, polished steel. This process yields an exceptionalfinish, but is expensive and makes the resulting shafting hard to work with. Inother applications, shafting with lower surface quality performs equally well andis cheaper and easier to process.

12.3.1 Rigid Shaft ConnectionsShafts are rigidly connected to other components in an assembly to provide sup-port, maintain (axial) position, or transmit torque. Here are some common types:

Left to right: A shaft collar with a split-hub clamp,a gear with a set screw, and a retaining ring.


• Split-hub clamps use force from a screw to apply pressure along the outsidesurface of a shaft. They transmit both axial and torsional loads well, have nobacklash, and tend not to loosen under cyclic loading. However, they requiremore space than other attachment methods. Simplified model analysis willshow that the peak axial force transmission is about Faxial ≈ 4 · µ ·Fs andpeak torque is about T ≈ 4 ·µ ·Fs · rshaft, where Fs is screw force.

• Set screws apply a normal force directly to the side of a shaft. They aresmall and simple to manufacture, but provide a weaker and less robust con-nection than split-hub clamps. The screw tends to be smaller, with a moredelicate drive feature, reducing screw force. The screw has no mechan-ical advantage, so T ≈ 2 · µ · Fs · rshaft. Set screws tend to loosen undercyclic loading, as small variations in the geometry of the tip of the screwand the shaft lead to counterclockwise torques during some loading condi-tions. With very little loosening rotation, the normal force goes to zero andthe connected part slips. Manufacturing a shallow, flat cut, or ‘flat’, on thesurface of the shaft improves robustness of the connection.

• Keyways and keys are slots machined into a shaft and mating part, andtight-fitting inserts into those slots, respectively. Keyways allow the highesttorque transmission, since interference between the key and keyway leads tonormal loading when torqued. Keyways are difficult to machine, however,especially for small shafts and hubs. Without very precise manufacturing,keyways exhibit backlash, in which the key is momentarily not in contactwith either side of the keyway during changes in direction of the drivingtorque, and small amounts of slipping occur.

• Retaining rings are thin metal arcs that fit into a mating groove to trans-mit axial loads. External retaining rings fit into grooves on a shaft, whileinternal retaining rings fit into grooves in the hole of the receiving part. Re-taining rings generally do not provide a useful function without this groove.They are installed or removed with specialized pliers. Retaining rings oth-erwise have similar strengths and weaknesses to keys as discussed above.

12.3.2 Shaft Selection ProcessTo select shafting:

1. Perform free-body-diagram analyses to estimate the bearing forces, torqueand peak bending moment for the shaft.

2. Select bearings to support the shaft (see above) and note the inner diameter.

12.4. GEARS 15

3. For one candidate material, such as hardened 440C stainless steel, deter-mine the minimum shaft diameters for (i) torque and (ii) bending moment.

4. If the bearing diameter is substantially larger than the other two diameters,try materials with lower strength and density. Iterate on step 3 until one ofthe load-based diameters is close to the bearing diameter.

5. If one of the load-based diameters is larger than the bearing diameter, trystronger materials in step 3 or select a larger bearing (this is rare).

6. Consult a catalog, such as McMaster-Carr, to find candidate stock. Tryconstraining material first and selecting diameter, which should match thebearing. Select a length sufficient to cut off your final shaft.

7. If something is off, for example other components that must mount to theshaft won’t fit, adjust assumptions and work through the process again.

12.4 GearsGears are used to constrain the angles of connected shafts or components, increaseor decrease torque or speed, and alter the speed-torque relationship across trans-missions. A common application is to convert the low-torque, high-speed outputof an electric motor into high-torque, low-speed output at a robotic joint.

In this section, we will describe gears qualitatively to stimulate your creativity.In a different chapter we analyze the mechanical basis for gear ratios, provideanalytical approximations of their strength and efficiency suitable for design, andprovide a process for their selection.

Left to right: Large, high-strength spur gears, high precisionspur gears with split-hub (fairloc) clamps, and spur gear stock.


12.4.1 Spur Gears

Spur gears are a common, basic type of gear, comprising a cylinder with teeth cutinto the outer surface. The teeth of two spur gears interlock, or mesh, such thatthe two gears rotate together in a constrained way. In selecting a spur gear set, themost important characteristic will be the gear ratio, which is the ratio of gear radiior, equivalently, number of teeth. In order for them to mesh, they must have thesame diametral pitch (different from diameter) and pressure angle. Meshing gearsshould generally be made of the same material and have the same face width.

12.4.2 Bevel Gears, Worm Gears and Rack and Pinion Gears

There are many other ways of orienting gear teeth to constrain movement:

Left to right: Steel helical bevel gears with a one to one ratio,plastic bevel gears with a higher ratio, and a steel worm gear.

• Helical gears have teeth oriented at an angle relative to the primary di-rection of load transfer, causing them to mesh more smoothly, improvingefficiency and reducing vibrations at high speeds.

• Bevel gears are conical, rather than cylindrical, allowing the axes of theconnected shafts to intersect at an angle, often 90◦. Bevel gears often requirestronger support structures, because their shafts must be cantilevered andsustain both radial and axial loads.

• Worm gears look and operate like a lead screw meshed with a spur gear,except for having teeth that mesh appropriately. They allow shaft axes to beoffset and oriented at an angle, often 90◦. Worm gears usually have a largegear ratio, and are usually non-backdrivable.

12.4. GEARS 17

• Rack and pinion gears have one round rotating gear, the pinion, meshedwith one straight linearly translating gear, the rack. They are similar to aspur gear set where one gear has infinite diameter.

Nearly any relative orientation of gears can be achieved with the correct toothshapes if needed. In some configurations, the meaning of ‘gear ratio’ can becomeconfusing. In some of these cases, the ratio can be obtained by comparing thenumber of rotations the input shaft needs to make to obtain one full rotation ofthe output shaft. In all cases, careful static analysis using free body diagrams willprovide an answer.

12.4.3 Gear IntegrationPrecise, stiff alignment is critical for gears to mesh correctly. If axes are too farapart, or are pushed apart due to compliance, then backlash, increased tooth mo-ments, skipping of teeth, or breakage can occur. If the axes are too close or notparallel, radial bearing loads, friction, and wear will be increased.

Reaction forces and moments from a gear onto its driving shaft include not onlythe transmitted torque, but also forces from tooth interactions. A spur gear withpitch radius r and driving torque T will have a radial force of about

Fradial ≈Tr

(12.7)

These loads must be accounted for when designing supporting structures such asshafts and bearings. Bevel gears will also have an axial reaction force and momentto be supported.

12.4.4 Gear Selection ProcessTo select a gear set:

1. Perform a motor set-point analysis and free-body-diagram analysis to deter-mine the desired gear ratio.

2. Consult a catalog, such as W.M. Berg or Stock Drive, to identify availablegear materials and sizes.

3. Use the selection guide in the Gears chapter to select a gear set with: thecorrect ratio; sufficient strength; low mass, high efficiency, low cost or acombination of these factors; and desirable mounting features.


4. Perform a free-body-diagram analysis to determine the forces and momentson the structure, such as a shaft, supporting the gear.

5. If something seems off, for example the gears or supporting loads are verylarge, adjust assumptions and work through the process again.

12.5 Chains and SprocketsChains and sprockets are similar to spur gears, except that the sprockets interactthrough a chain rather than meshing directly. This has the advantage that smallsprockets can be located at a distance, provided they are in the same plane.

Chains are rated for driving tension, equivalent to the tooth force in spur gears,equal to the driving torque divided by the pitch radius. The peak driving tensiondepends on chain material and size. Exceeding the driving tension will shear thechain links. Chains must have the correct length to wrap around the sprockets, orthey can skip teeth.

Two examples of flex-e-chains and accompanying sprockets.

12.5.1 Chain Selection ProcessTo select a chain and sprocket set:

1. Perform a motor set-point analysis and free-body-diagram analysis to deter-mine the desired gear ratio and torques.

2. Consult a catalog, such as W.M. Berg or Stock Drive, to identify availablechain types and sprocket sizes.

3. For one candidate chain type, calculate the required sprocket diametersbased on the rated operating load and required torques.

4. Perform a free-body-diagram analysis to determine the forces and momentson the structure, such as shafts, supporting the timing sprockets.

12.6. TIMING BELTS AND PULLEYS 19

5. If something seems off, for example the sprockets or supporting loads arevery large, adjust assumptions and work through the process again.

6. Select the desired sprockets and a chain that has exactly the correct lengthfor the supporting shaft locations.

12.6 Timing Belts and Pulleys

Timing belts and pulleys are similar to spur gears, except that the pulleys interactthrough a belt rather than meshing directly. This has the advantage that smallpulleys can be located at a distance. Some belts will tolerate bending along thebelt axis, but timing belts work best if the two pulleys lie in the same plane.

A variety of timing pulleys, timing belts, and idlers.

Example of a timing belt drive with idler and tensioner pulleys.


Timing belts are rated for driving tension, equivalent to the tooth force in spurgears, equal to the driving torque divided by the pitch radius. The peak drivingtension is typically linearly proportional to belt width. Exceeding the driving ten-sion will cause the polymer teeth of the belt to slip past the teeth in the pulley.

Timing belts require some amount of pretensioning to prevent slippage as theystretch under load. Pretensioning can be achieved through residual loads appliedduring assembly, e.g. forcing a supporting component sideways and then tighten-ing it down, or through an additional idler pulley that applies a force perpendicularto the belt in an unsupported region. An idler can also help take up slack arisingfrom imprecise mounting. Failure to pretension will result in belt skipping.

12.6.1 Timing Belt Selection ProcessTo select a timing belt and timing pulley set:

1. Perform a motor set-point analysis and free-body-diagram analysis to deter-mine the desired gear ratio and torques.

2. Consult a catalog, such as W.M. Berg or Stock Drive, to identify availablebelt types and timing pulley sizes.

3. For one candidate timing pulley and belt type, calculate the required pulleydiameters based on the rated operating load and required torques.

4. Perform a free-body-diagram analysis to determine the forces and momentson the structure, such as shafts, supporting the timing pulleys.

5. If something seems off, for example the pulleys or supporting loads are verylarge, adjust assumptions and work through the process again.

6. Develop a mechanism for pretensioning the timing belt.

7. Select the desired timing pulleys and a belt that has exactly the correctlength for the supporting shaft locations.

12.7 Cables, Capstans and PulleysCable, or wire rope, can be used to provide static tensile forces or to transmitpower. When used for power transmission, the elements onto which the ropewraps are called capstans or drums, elements for rerouting rope are pulleys, andthe overall design is a cable drive transmission.

12.7. CABLES, CAPSTANS AND PULLEYS 21

Stainless steel cable (left), vectran rope (center left), capstan (center right),pulley with internal bearing (right), and a variety of cable constructions (bottom).

The most important properties to consider when selecting rope are breaking strengthand flexibility. Breaking strength is most strongly affected by material and diame-ter. Strength is measured under ideal conditions and should always be used with alarge factor of safety, especially if the rope is cyclically wrapped and unwrapped.Flexibility of a cable is determined by its material and its construction, or theweave of the individual fibers. Small cables with many small fibers and weavestend to result in more flexible construction, but lower strength and higher cost.

Common cable materials include stainless steel and polymers like Vectran. Stain-less steel is stiff and strong, but tends to be relatively resistant to bending and issusceptible to abrasion, requiring larger diameters in well-designed cable drives.As a heuristic, the minimum capstan or pulley diameter is 20 times the diameterof the wire rope. Coatings help prevent wear and improve gripping for wire ropes.

Vectran is strong, flexible and abrasion resistant, and has low density, making it anexcellent material in many cable drives. However, it also exhibits more stretch, orelongation under load, and creep, or permanent changes in nominal length, thansteel ropes, making it less desirable for high-precision drives such as in hapticinterfaces. Vectran can be wound on drums as small as 3 times the rope diame-ter and will hold knots. Nylon, monofilament fishing line and natural-fiber ropeusually have poor performance in robotics applications and are not recommended.

When integrating cable into a mechanical system, note that wrapping the cablewill lead to friction. This can help; wrapping the cable a few times before tyingit will greatly reduce force on the knot, a weak point. It can also hurt; routing the


cable through an eyelet will lead to friction power losses, which can be very large.We will analyze these in a future chapter. If you are using a cable as part of apower transmission, and the cable must turn a corner between drums, it is highlylikely that a pulley on a proper bearing will be necessary for good performance.

12.7.1 Cable Selection Process

To select cables:

1. Perform a free-body-diagram analysis to estimate cable tension.

2. Use the heuristics above to guess at an appropriate cable type.

3. Consult a catalog, such as McMaster-Carr or West Marine, to find candi-date cables. First constrain material, then breaking force (accounting forfos). Finally choose a construction that reflects whether the cable will bend(many fibers) or not (fewer fibers). Select an appropriate length.

4. If the design involves drums and pulleys, use the above heuristics to set theirdiameters as a multiple of cable diameter.

5. If something seems off, for example a diameter is very small or large, try analternate material or adjust assumptions and iterate as needed.

12.8 Springs

Springs provide force as a function of displacement, which can be used to providepassive loading, store energy, or even to add a degree of freedom as a flexure. Themost important properties in spring selection are peak force and stiffness or dis-placement.

12.8. SPRINGS 23

Examples of compression springs (left), tension springs (upper right),and torsion springs (top center and lower right).

Minimum spring size is strongly influenced by the energy stored, spring geometryand material properties. The most efficient spring geometry is a prismatic memberin axial load, since each element experiences equal (maximal) stress and strain,but most engineering materials are too stiff to make this layout practical. Helicalcoil springs are 3 times heavier than this standard, because the wire is loaded intorsion, and are made from steel which has relatively poor energy storage density(see below), but are compact and widely available. Leaf springs are 9 times heav-ier than the axial standard, because they experience cantilever bending loads, butcan be constructed from composite materials with superior energy storage density.

For most spring geometries, the ideal spring material will minimize ρ ·E · S−2y ,

where ρ is density, E is elastic modulus, and Sy is yield strength. Best in class ma-terials include unidirectional fiberglass used in leaf springs, ρ ·E ·S−2

y ≈ 6×10−5,music wire (hardened steel) used in helical coil springs, ρ ·E ·S−2

y ≈ 5×10−4, andrubbers used in axial springs, ρ ·E ·S−2

y ≈ 5×10−5.

12.8.1 Spring Selection Process

To select steel coil springs:

1. Determine the required peak spring force and displacement. In some cases,this will be obtained through static balance and geometry. In other cases, itmay be determined by energy storage requirements, where E = 1/2 ·F ·∆xfor linear springs. Note the implied stiffness, k = F/∆x.

2. Consult a catalog, such as McMaster-Carr, to find a candidate spring. First


constrain by force, then by displacement or stiffness. Note the overall di-mensions of the resulting spring.

3. Repeat step 3 a few times, choosing slightly different combinations of forceand stiffness near the desired values to see what is possible.

4. If all of the capable springs are very large, you may choose to make designmodifications to reduce the force or displacement requirements, or fit a largespring, and work through the process again.

To design axial rubber springs or fiberglass leaf springs, use simplified models ofstress and displacement to select optimal geometric parameters, such as length,cross-sectional area, width or height, using the inverse-analysis methods detailedin Chapter 4. Rubber springs can be modeled as members in axial loading, andleaf springs can be modeled as beams in cantilever bending.

12.9 Electric MotorsElectric motors convert electrical power into mechanical power. More precisely, amotor converts electrical current into mechanical torque, and experiences a volt-age drop in proportion to the current, due to coil resistance, and a current dropin proportion to how fast it is spinning. Understanding electric motors requiresadditional fundamental knowledge, which we address in a separate chapter.

A variety of electric motors, windings and gearboxes, from Maxon Motor inc.

12.10. BATTERIES 25

To select an electric motor:

1. Determine desirable power and torque capabilities for your design scenariousing appropriate analyses.

2. Consult a catalog, like Maxon Motor, Pololu or Kollmorgen, to identify acandidate motor. Try constraining by rated power first.

3. Use the techniques from the Motors chapter to select an appropriate operat-ing point for the motor. This will relate to desired power or efficiency, andto limits on voltage, speed or heat production.

4. Calculate the desired gear ratio as the ratio of the desired torque to the motortorque at this operating point. Select a gearbox that is compatible with themotor from the same catalog.

5. Update the motor analysis given the selected gearbox efficiency. Iterate ongearbox selection as necessary.

6. Select a battery that can achieve the required voltage and current.

7. Note the mass, size and cost of the chosen motor, gearbox and battery.

8. Go through this process iteratively, choosing a different candidate motoreach time. At the beginning of the process, try very different types of motorto see what classes seem appropriate. Later in the process, try differentmodels within product lines that seem effective.

12.10 BatteriesBatteries are the most common source of electrical power for mobile devices, in-cluding robots. We use them constantly, but probably haven’t thought much abouttheir properties. Here are the key characteristics for design purposes:

A low-C LiPo battery, high-C LiPo battery, and NiCd battery.


• Battery Cell. A battery is comprised of ‘cells’, in which a chemical reac-tion occurs. The chemical reaction converts stored chemical energy intoelectrical energy (or the reverse when charging). The materials involved inthe reaction have a strong effect on battery properties. Batteries are usuallyfirst categorized by this chemistry, for example LiPo. Cells can be placed inseries or parallel, but cannot be divided.

• Voltage (V). This is the electrical potential, or electromotive force, thatpushes electrons from the anode to the cathode. The Voltage of a singlebattery cell is set by its chemistry and represents the minimum nominal volt-age. Higher voltage, in integer multiples of cell voltage, can be achieved byplacing cells or batteries in series. If the current draw goes beyond the ratedvalue (see below), the voltage can drop well below its nominal value. Forexample, if you short a battery the voltage will go to zero.

• Energy Storage (Ah). This is the total energy available from the battery ona single charge. It is typically given in units of Amp Hours, the integralof current over time, corresponding to the number of hours for which thebattery could output 1 Ampere of current (if this were within its allowabledischarge rate). To convert to Joules, multiply the Voltage by the capacityin Ah, and multiply the result by seconds per hour (3,600).

• Maximum Discharge Rate (C). The current, or electrons per second, thatcan be generated by a battery cell will depend on its size (reacting surfacearea). The maximum current is usually given as a discharge rate, in unitsof ‘fractions of the battery capacity’, or C. To convert to units of Amperes,multiply the capacity in Ah by the discharge factor, C. Even within a givenbattery chemistry and capacity, a wide range of values of C are possible.Peak discharge rate can also be increased by placing batteries in parallel.Peak current is often a critical factor in battery selection for robots.

Many battery types are available, but a few are commonly useful in mechanicalsystems design:

• Lithium Polymer (LiPo). Often a good choice for high-performance roboticsystems. Cell voltage is 3.7 V. High energy storage density (0.7 MJ/kg).High discharge rates possible (1–50 C). High cost (1.2 $/Wh). Recharge-able, but can explode so always use a LiPo-specific charger.

• Alkaline (AA, 9 V). Often good for simple consumer products. Cell volt-age is 1.5 V. Medium energy storage density (0.5 MJ/kg). Typical peakdischarge rates of 1 C. Medium cost (0.1 $/Wh). Not usually rechargeable.

12.11. PROMINENT CATALOG SOURCES 27

• Ni-Cad and Nickel Metal Hydride (NiCd, NiMH). Good rechargeable choicefor simple consumer products. Cell voltage is 1.2 V. Medium energy stor-age density (0.4 MJ/kg). Typical peak discharge rates of 1 C. Medium cost(0.2 $/Wh).

• Lead-acid (PbSO). Good for cheap, heavy, high-capacity energy storage.Cell voltage is 2.0 V. Low energy storage density (0.1 MJ/kg). Typical peakdischarge rates of 1 C. Low cost (0.01 $/Wh).

How to select a battery

1. Before selecting a battery, design of the electrical load to be driven, forexample the electric motor and transmission, noting the required run time.This will set the desired voltage, current and energy storage.

2. Calculate the required discharge rate factor (in C) by dividing the peak cur-rent (in A) by the needed energy storage (in Ah).

3. Consult a catalog, such as SparkFun or Amazon, to find a candidate bat-tery. Select a battery type using the heuristics above, then find a part withapproximately the correct energy storage, discharge rate and voltage.

4. If the available battery voltages are too low, see if you can achieve approx-imately the correct voltage by placing multiple batteries in series. If thebattery discharge rate is too low, see how many batteries you would need toplace in parallel to achieve at least that rate.

5. Now return to the motor design problem to see whether the available batteryproperties will achieve adequate motor performance. Check also what thetotal mass and cost of the battery or batteries would be.

6. If things are way off, use your new knowledge of the available batteries toguide iterative re-selection of the motor, operating point and battery.

12.11 Prominent Catalog Sources

There are many companies that sell catalog components, and many more still thatspecialize in supplying a particular type of component to both catalogs and indus-try. Here is a sample related to course projects, research, and initial prototypes,including some good places to look first:


• McMaster-Carr, http://www.mcmaster.com, is a good place to lookfor fasteners, bearings, shafting, cable, and springs. They offer a wide va-riety of other machine components, with a bias towards larger applications.McMaster has fast shipping and a user-friendly website. Watch out for ship-ping costs, which are not predicted during checkout.

• Stock Drive Products, or Sterling Instruments, http://www.sdp-si.com, is a good place to look for small-scale gears, timing belts, pulleys,chains and sprockets. Their web interface is also reasonably user friendly.

• W. M. Berg, http://www.wmberg.com, has similar offerings as StockDrive.

• Quality Transmission, http://www.qtgears.com, has similar offer-ings as Stock Drive.

• Dragon Plate, http://www.dragonplate.com, sells a variety of car-bon fiber stock and offers custom machining.

• Carbon Fiber Tube Shop, http://www.carbonfibertubeshop.com, sells... carbon fiber tubes.

• Gordon Composites, http://www.gordoncomposites.com, offershigh-quality composite materials well-suited to leaf springs.

• West Marine, http://www.westmarine.com, sells synthetic ropessuch as Vectran. And sailing equipment, if that’s handy.

• Jameco and Mouser, http://www.jameco.com and http://www.mouser.com, sell a variety of hobbyist electronic components.

• Digi-Key, http://www.digikey.com, sells a wide variety of elec-tronics hardware.

• Sparkfun, http://www.sparkfun.com, sells hobbyist robotics com-ponents.

• Amazon, http://www.amazon.com, sells just about everything, if youknow what you’re looking for, although most robotics hardware here is notwell described.

12.12 AcknowledgmentsThanks to Kirby Witte for help in editing this chapter.

http://www.mcmaster.com

http://www.sdp-si.com

http://www.sdp-si.com

http://www.wmberg.com

http://www.qtgears.com

http://www.dragonplate.com

http://www.carbonfibertubeshop.com

http://www.carbonfibertubeshop.com

http://www.gordoncomposites.com

http://www.westmarine.com

http://www.jameco.com

http://www.mouser.com

http://www.mouser.com

http://www.digikey.com

http://www.sparkfun.com

http://www.amazon.com

Bibliography

Richard Budynas and Keith Nisbett. Shigley’s Mechanical Engineering Design.McGraw-Hill, 8th edition, 2006. ISBN 0-390-76487-6.

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