# Rigidty: Transmission Design Considerations

## Transmission Design Considerations of Cam Mechanisms

Input Transmission: all of the transmission components from the power source (usually an electric motor) to the cam.

Output Transmission: all of the transmission components from the cam-follower (usually a roller) to the payload (also called: end-effector, or tool).

The components in the transmissions typically include shafts, gears, gearboxes, couplings, chain drives, belt drives, linkages.

### Three[3] Mechanical Properties of the Transmission:

 The performance of the input and output transmissions is a function of three parameters: Strength... the ability to withstand the forces and torques without fracture or yield. The design must be strong enough to transfer the peak force or torque. The design of components is more in the field of strength of materials, not cam systems. Rigidity... the ability to transmit the force and torque without too much deflection. Rigidity is important in the generation of vibration. All transmission components have elasticity, the reciprocal of rigidity. When a metal component is stressed within its elastic limit, it strains elastically. Its distortion, or deflection, is related to its size and shape, and is proportional to the load applied. When it is stressed beyond its elastic limit it suffers plastic deformation. Plastic deformation does not recover when you remove the stress. It may also suffer hysteresis within the elastic limit. Hysteresis is an energy absorbing phenomenon, whereby the strain produced by an increasing load is not fully reduced by a decreasing load. Hysteresis is responsible for internal damping of vibrations. But this effect is unlikely to be significant in cam systems. To simplify analysis, we assume that transmission components are perfectly elastic with no hysteresis. If there are several components connected in series, as is a typical cam transmission, the deflections add together so that the overall deflection from one end of the transmission to the other is the sum of the individual deflections. When gearing is involved, different parts of the transmission may be subject to different torques and the deflections at one side of the a gear pair may be transformed in a different deflection at the other side of the gear. To assess the rigidity of a transmission as whole, therefore, it is necessary to estimate the rigidity of each component and combine them in a particular way. Backlash...lost transmission with a reversal of torque or force. We will review the detrimental effects of backlash in the next topic. Here.

### Input Transmission: Rigidity and Stiffness

The Input Transmission includes the power transmission components from the power source [usually a motor] to the cam.

Most cams in industrial machines rotate and they are driven by rotating motors. Thus, the components are rotary, and they will have an angular deflection that is proportional to the applied torque (pulleys and sprockets of belts and chains are also 'rotary components'). The simplest components are shafts, but these are often constructed with sections of different diameters. Shafts that are connected in series frequently have different diameters.

#### Shaft Rigidity

The Rigidity of rotary components is defined as the torque divided by the angular deflection, [T/Θ = G.J/L] :

$R=\frac{\pi .G\left({D}^{4}-{d}^{4}\right)}{32.L}$ ....Equation 1

R  = Rigidity of Circular Shaft [N.m/rad]

G  = Modulus of Rigidity of the Material (Shear Stress/Shear Strain) [N/m2]

D  = Outside diameter of the Shaft [m]

d   = inside diameter of shaft [m]

L   = Length of shaft subject to torsion [m]

If you want the result in 'English/American' units, I always do all calculations using Metric units, and find a conversion table to convert the answer to English units.

Note: I do not know any British/English engineer, that is younger than 90years old, who works with 'English' units.

The overall Rigidity of a shaft with three sections, calculated using the above rigidity equation is:

$\frac{1}{R}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+\frac{1}{{R}_{3}}$ ....   Equation 2

This equation applies to any mixture of diverse components - gears, couplings - that are connected in series.

From this equation, it can be seen that the overall rigidity of the complete transmission is always less than the rigidity of any of its components.

The least rigid component is the most significant. Often the least rigid component dominates the result, such that very rigid components are not important to the final result.

EXAMPLE: A Stepped Shaft: three different lengths and diameters in series.

 D1 = 30mm d1 = 15mm L1 = 40mm D2 = 25mm d2 = 15mm L2 = 35mm D3 = 22mm d3 = 0mm L3 = 160mm

[Modulus of Rigidity of Steel, G = 82.5x109 N/m2]

1/R = 0.000006503 + 0.00001271 + 0.00008433 = 0.00010354

This shows that the overall rigidity is less, but not much less, than the least rigid part of the shaft. Had we ignored the most rigid part - R1 - the result would have been 10,305 and not 9,657N.m/rad, only 7% more.

#### Gear Rigidity

 There are several type of gears used in industrial machines - Spur Gears, Worm Gears and Belt and Chain Drives. Generally, the gear wheel itself has a torsional rigidity that is high enough to be ignored, but the loaded gear teeth themselves may distort sufficiently to contribute significantly to the overall elasticity of the transmission, particularly on small diameter pinions. Both the driving and the driven teeth bend and also compress under Hertzian Stress. These two distortions combine to give tangential linear deflection at the pitch line which varies somewhat as each tooth passes through the contact zone. But this variation can be ignored here. The stiffness* of the the tooth can be defined as the tangential force divided by the tangential deflection when the contact point is on the common center-line, and can only be approximately calculated from the tooth dimension and material properties. When possible it is best to measure the torsional rigidity rather than estimate it. However, some design guidance is derived from considering the relationship between tooth stiffness and torsional rigidity*. * In this topic, we use Stiffness to apply to Linear-Stiffness (N/mm), and Rigidity to apply to Angular-Stiffness (N/rad). The image shows a schematic of 'rigidity* of a spur-gear' - a pair of unequal spur gears is transmitting a torque with a tangential tooth force. The deflections are analogous to those of a pair of levers whose tips are connected by a spring, as shown as the 'Equivalent Gear-Pair'. Relating Force, Stiffness and Deflection Total Tangential Linear Deflection, δ $\delta =F/S$   F Tangential Force [N], at the pitch-circle [equal and opposite of course] S : Linear Stiffness [N/m] of gear teeth in series [F/δ] Thus, for small angles, angular deflections: Gear A rotates by of δ/ra radians, if Gear B is held stationary Gear B rotates by δ/rb radians, if Gear A is held stationary. Relating Torques, Stiffness and Deflection Let Ma and Mb be the Moments [Torques applied] on Gears A and B , whose pitch circle radii are ra and rb. [m]. The torsional rigidity of Gear A relative to Gear B is Ra,(= Torque[N.m] ÷ angular deflection[rad] ), and is found with: ${M}_{a}=F.{r}_{a}$ ∴ ${R}_{a}=\frac{{M}_{a}}{\delta /{r}_{a}}=\frac{{F.{r}_{a}}^{2}}{\delta }={S.{r}_{a}}^{2}$   ... Equation 3 Similarly, the effective rigidity of Gear B relative to Gear A is ∴ ${R}_{b}=S.{{r}_{b}}^{2}$   ... Equation 4 In general, Torsional Rigidity, R, is related to Linear Stiffness, S, acting at a radius, r, by the equation: ∴ $R=S.{r}^{2}$   ... Equation 5 In a well designed transmission system, the linear stiffness of the gear teeth is not very important. The torsional rigidity of a gear is proportional to the linear stiffness of the gear tooth. Also, the larger the tooth the more rigid the gear. More important is that the Torsional Rigidity is proportional to the square of the pitch circle radius. Therefore, use large gears if possible, even if small gears are strong enough. Similar results are obtained from the analysis of the rigidity of other types of gear pair, such as bevel gears or worm gears. The estimate of tooth stiffness from design information may not be very accurate. Therefore, it is best to obtain a measured rigidity value from the gear manufacturer, or from a bench test. All types of gearing usually operate with a small amount of backlash. This can cause problems in a cam transmission when when there is a reversal of torque in every motion cycle. The aim in cam transmissions is to have minimum backlash, with an acceptable initial cost. Note, the power-loss may increase with increased friction as backlash is reduced and cause high-speed drives to overheat. As we have already seen, the overall elasticity of a power transmission, is the sum of the elasticities of each section when connected in series (Equation 2). In effect, the elasticity of one section is 'transmitted' to the next. When there is gearing between the two sections, however, the transmitted elasticity is modified by the gear ratio. To study this effect, assume a pair of gears with infinitely stiff teeth, the input gear having Zi teeth, and the output gear with Zo teeth. The gear ratio is: ${Z}_{i}/{Z}_{i}$. Now, let: Ri = Rigidity of all mechanical components before the output gear R0 = Rigidity of all the mechanical components after the gear. also: Mi = Input Torque of the Gear Pair Mo = Output Torque of the Gear Pair The torsional deflection of the input shaft is: ${M}_{i}/{R}_{i}$ This is transmitted as a deflection at the output gear as $\left(\frac{{M}_{i}}{{R}_{i}}\right).\left(\frac{{Z}_{i}}{{Z}_{o}}\right)$ The torsional deflection of the output sections of the transmission is ${M}_{o}∕{R}_{o}$ This is added to the transmitted deflection. The total deflection at the output end of the transmission is therefore: ${M}_{0}∕{R}_{o}$ The overall elasticity, (reciprocal of Rigidity) as seen at the output is this deflection divided by the output torque: $\frac{1}{R}=\frac{{M}_{i}.{Z}_{i}}{{R}_{i}.{Z}_{o}.{M}_{o}}+\frac{1}{{R}_{o}}$ Ignoring Gear Efficiency, $\frac{{M}_{i}}{{M}_{o}}=\frac{{Z}_{i}}{{Z}_{0}}$   Therefore, the equation for overall elasticity at the output becomes: $\frac{1}{R}=\frac{1}{{R}_{i}{\left({Z}_{o}∕Zi\right)}^{2}}+\frac{1}{{R}_{o}}$ ...Equation 6. This is similar to Equation 2, but that the first term has been modified. The rigidity 'transmitted' by gearing is multiplied by the gear ratio squared. In a similar way, we can show that the backlash 'transmitted' by gearing is directly proportional to the gear ratio. A reduction gear increases rigidity. A step-up gear reduces rigidity. Thus, when a long transmissions is unavoidable and a reduction gearing is necessary, make the longest part of a geared transmission be the high-speed shaft:  it transmits less torque than the low-speed shaft, and thus a smaller diameter based on strength, and its elasticity is reduced by the square of the gear ratio.

#### Chains and Belts Rigidity

 Equations 3, 4 and 5 can  be applied to chain drives, but in that case the stiffness, S , refers to the stiffness of the loaded length of chain between the chain-wheels. For a given chain size, the stiffness is inversely proportional to its length: very long chain drives should therefore be avoided as should small diameter sprockets. Belt drives behave in a similar way, but are generally less satisfactory than chain drives. Flat belt and vee belt drives are seldom used in cam system transmissions (except at very high speed, e.g. the primary drive from the electric motor where their elasticity is not important). Timing belt drives [belts and pulleys with 'teeth'] are common because they give an exact speed ratio for synchronizing with other mechanisms in the machine. Timing belts are made of reinforced synthetic rubber and are rather elastic compared to metal chains of similar strength. This is partly because the rubber belt teeth tend to roll slightly in the pulley grooves under heavy loads. More recently, the tooth profile is improved so that this problem is reduced. Nevertheless, timing belt drives are very successfully used in cam transmissions because they are almost silent and need no lubrication. The backlash problem with chains and belts is similar to that with gears, but usually more severe. Slack chain drives are quite common in conventional steady torque transmissions, and not particularly detrimental to them. The use of chain tensioner devices, of which there are many types commercially available, is strongly recommended for all cam transmissions, and are essential for high-speed or high-inertia applications.

#### Bearing Support Rigidity

 One effect of using gears, chain drives, etc. in a transmission, which is often overlooked, is the flexibility of the bearing supports. The tooth load produces an equal reaction force at the gear supports. When the gears are in a rigid casting (for example, a commercial gear-box and cam-box) the elastic deflections of the supports are usually small enough to be ignored. However, the reaction torque on the structure that supports the cam-box may itself be important. A rigid gear-box or cam-box is no advantage if it is not rigidly supported, or the frame deflects. Gears and chain wheels are sometimes unavoidably mounted on shafts far from the shaft bearings. This means the bending of the shaft due to the tooth load becomes a significant part of the overall rigidity of the transmission. Lateral deflection of the shaft has the same effect on angular displacement as tooth deflection and is mechanically in series with it. The image shows how the linear deflection of the shaft produces an angular deflection of the gear (or chain wheel) so that the effect is similar to torsional elasticity. Here the shaft is displaced by the tooth contact force, F.sec(Ф), where F is the tangential force and Ф is the gear pressure angle. This acts at a distance of r.cos(Ф) from the centre of the shaft, Therefore, the shaft torque is: $F.\mathrm{sec}\Phi .r.\mathrm{cos}\Phi =F.r$ The angular deflection of the shaft, however, is $\left(\delta .\mathrm{cos}\Phi \right)∕r$ Thus, the effective torsional rigidity is: $R=\frac{F.r}{\left(\delta .\mathrm{cos}\Phi \right)∕r}=\frac{F.{r}^{2}}{\delta .\mathrm{cos}\Phi }$    .... Equation 7 ...where δ is the lateral [sideways] deflection of the shaft. If the lateral stiffness of the shaft in the plane of the gear is S, then: $S=\text{force╱deflection}=F.\mathrm{sec}\Phi /\delta$ ...and the equivalent Torsional Rigidity is... $R=S.{r}^{2}$ This is the same as Equation 2. The relationship between lateral shaft stiffness and torsional rigidity is exactly the same as for tooth stiffness and is independent of the gear tooth pressure angle. From beam bending theory, we find that for a simply supported shaft of constant cross-section, the lateral deflection at the sprocket for a unit load is: $\delta =\frac{{{l}_{1}}^{2.}{{l}_{2}}^{2}}{3.E.J.L}$   ...and its reciprocal is $S=\frac{3.E.J.L}{{{l}_{1}}^{2.}{{l}_{2}}^{2}}$     .... Equation 8 Where: S = lateral stiffness of the shaft [N/m] l1 & l2 = distance from each side of the sprocket to each bearing that supports the shaft [m] E = Young's Modulus of elasticity of the shaft material J = second moment of area of the shaft cross-section. For bending of a circular shaft, J = Π. (D4 - d4)/64 [m4] L = Total length of shaft between the supports [m] If the shaft bearings are themselves mounted on a flexible frame structure, the lateral deflection of the frame produced by bearing reaction forces must also be taken into account in a similar way to the lateral deflection of the shaft. Structural flexibility is in series with all the other transmission elasticity.

#### EXAMPLE: Input Transmission Rigidity

Design Arrangement of an Input Transmission - From the Drive Motor to the Cam.

A Camshaft is driven by a 2:1 reducing chain drive from a primary shaft, which is driven by a motor and worm gear-box.

Find the overall rigidity of the input transmission to the cam, from the output of the worm gear-box.

Note: We want to find out how effectively the transmission is given to the cam by the worm gear-box.

The shafts are made of medium carbon steel, and the chain and solid sprockets are steel. G = 82.5 x 109 N/m2

The coupling is sufficiently rigid in torsion, and the shaft mountings are stiff enough to be ignored in the calculation.

The transmission chain is 0.5inch pitch with a stiffness of 6 x 106N/m, for a 1m length.

Find the rigidity of each section of the transmission separately and then combine them into one overall rigidity.

1: Rigidity of the Primary Shaft: Torsion

The section of the shaft transmitting torque is Ø38, and 454mm long. Therefore its rigidity is:

This shaft is connected to the cam-shaft by a 2:1 reduction drive. Therefore, its rigidity referred to the cam-shaft is:

2: Rigidity of the Primary Shaft: Bending

The Pitch Circle Radius of the Sprocket is 61mm.

Therefore, the equivalent torsional rigidity is: (using Equation 3)

The Rigidity referred to the cam-shaft via the 2:1 reduction is:

3: Rigidity of the Drive Chain

From the drawing above:

Length of chain that stretches under load = 496mm.

Stiffness, for 1m length  = 6 x106 N/m

The Pitch Circle Radius of the Chain-wheel is 122mm. Therefore, the rigidity of the chain referred to the cam-shaft is:

4: Rigidity of the Cam Shaft Bending

5: Rigidity of the Cam Shaft Torsion

Overall Torsional Rigidity of the Input Transmission

The Overall Rigidity of the transmission referred to the cam is:

Comment:

The most significant element with elasticity is the bending of the primary shaft. It has the lowest Rigidity, R2. This points to the possibility of a considerable improvement if:

 • The chain-drive could be moved closer to the right-hand bearings

and / or

 • the sprocket could be increased in diameter

### Output Transmission

The estimation of rigidity of an output transmission is exactly the same as for an input transmission. Most inputs, of course, drive rotary cams and therefore rigidity is expressed as the overall torsional rigidity. With output transmissions, however, we are dealing with a payload that is driven by the cam follower and the motion may be linear (reciprocating) or rotary (oscillating or indexing).

If the follower motion is a translating, linear motion, then the overall rigidity of the output transmission is expressed as a linear stiffness referred to the follower.

If the follower motion is a swinging, rotating motion, then the overall rigidity of the output transmission is expressed as a torsional rigidity referred to the follower axis.

 Levers and links are similar to gears and chain-wheels. A linear deflection, δ , at a point under Force, F, at a distance, r , from the lever pivot, or fulcrum, can be expressed as a Linear-Stiffness S = F/δ at that Point. This is translated into Torsional-Rigidity at the pivot of R=S.r2. This is the same as Equation 5, above. The designs of levers are many and varied. The calculation of lever stiffness is therefore not considered here. It comes within the conventional theory of deflection of beams. In practice, well designed levers are seldom a significant source of elasticity in transmissions, unless they are very long. The elongation and compression of link (pull or push-rods) are analogous of a chain, described above, and the relationship between the stiffness of a link and the torsional rigidity at a lever pivot is exactly the same as between a chain and its chain-wheel shaft. Equation 5 applies.

#### EXAMPLE: Output Transmission Rigidity

 Design Arrangement of an Output-Transmission - from the cam-follower to the payload. A cam-driven mechanism is operated by a lever and pull-rod transmission with a stroke-increasing ratio of 1 .5:1 as shown in Fig. 1 1 .5. A 60 mm long follower arm is keyed to one end of a 20 mm diameter steel pivot shaft on the other end of which is keyed a 90 mm long pull-rod lever. That lever pulls a payload by means of an 8 mm diameter x 120 mm long steel pull-rod. Find the overall transmission rigidity from the cam to the payload, assuming that the follower arm and pull-rod lever are stiff enough to be ignored in the calculation. Working back from the payload to the cam: 1: Pull-Rod The stiffness of a bar in pure tension or compression is: where: E = Young's Modulus (~205 x 109 N/m2 for steel) A = cross-section area L = length under stress This acts at the end of a 90mm long lever. Thus, the torsional rigidity referred to the pivot shaft (Equation 5) is: 2: Pivot shaft bending at pull-rod position Using Equation 8 and 5: Rigidity referred to the pivot shaft is: 3: Pivot shaft torsion The length of the shaft in torsion is 240mm. From Equation 1 4: Pivot shaft torsion The shaft is symmetrical along its length and its stiffness in bending, S , at this position is the same as at the pull-rod position. The rigidity referred to the pivot shaft is therefore: The overall torsional rigidity of the cam output transmission at the follower arm pivot pivot is: This could be expressed as a linear stiffness at the follower roller by transposing Equation 5: It is clear from the above figures that torsion of the pivot shaft is by far the most elastic element, showing that the transmission rigidity can be considerably improved, if necessary, by increasing the shaft diameter.  Because shaft rigidity and stiffness are proportional to the fourth power of diameter, an increase from 20mm to 24mm would approximately double the overall rigidity.

#### Couplings

Couplings are important in any transmission.

They must be able to compensate for misalignment between two shafts.

The misalignment may be classified as:

 • Parallel Offset
 • Angular
 • Axial

Coupling may also prevent a transmission shock to be transmitted. They may be designed to de-couple if a torque is exceeded.

There are many types and designs of shaft coupling that are available commercially, as well as those for special purpose designs.

Rigid

'Rigid' they are not intended to allow relative movement of any kind between the coupled shafts.

Self-Aligning

'Self-aligning' allow limited movement between the shafts, usually to provide for either accidental or deliberate misalignment.

Of the latter, some allow all degrees-of-freedom - these are the 'flexible' couplings - and some allow only one or two degrees-of-freedom.

Flexible couplings transmit torque via a resilient medium (rubber, plastics or metal spring) and are not torsionally rigid. They are not recommended for cam systems.

Torsionally Rigid

Of the torsionally rigid couplings, the:

Cardan Joint allows a large degree of angular misalignment

Oldham Coupling allows a large degree of parallel offset and some end float.

Gear and the Chain Couplings allow a small degree of freedom in all directions except torsion.

With the exception of the membrane type, the mechanical couplings are subject to gradual wear which results in rotary backlash, which may be a problem for cam drive transmission if the couplings are not large enough.

Splined Shaft Coupling is a form of coupling which allows substantial axial displacement of a shaft while retaining torsional rigidity. However, it must have clearance to allow for the relative movement, and because the pitch circle radius of splines is inevitably small the rotary backlash is potentially severe.

Membrane or Diaphragm Couplings have a thin flexible membrane, sometimes laminated, attached alternately to the two hubs.

This allows limited angular misalignment and end float, but virtually no torsional elasticity and no backlash. Lateral shaft offset, if required, can be achieved using two such couplings separated by a short length of shaft.  This design of coupling is ideal for cam drives, for both input and output transmissions.

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