EngineeringMechanical EngineeringAppendix to the series
Machine Element Design by Proportion and Analysis
If it looks right it probably is right. That old workshop principle is a genuine design method — provided you know why the proportions work and can prove it when they do not.
- Good proportions
- Complete stress analysis
- Cast vs fabricated
- Three worked elements
Executive summary
The preceding parts of this series dealt with elements that are bought in. This closing part deals with elements that are made — and demonstrates the complete design process on three of them: the rigid coupling, the knuckle joint and the lever.
Design proceeds on two tracks at once. Established proportions, expressed as simple functions of a governing dimension, produce a shape that is known to work. Strength of materials analysis then confirms — or corrects — that shape. Neither track alone is sufficient: proportions fail when a design carries unusual materials or constraints, and analysis alone produces geometry no one would recognise as a coupling.
Manufacturing method drives the geometry
Casting
Suited to production quantities. Allows generous fillets, webs and varying section thickness at no extra cost per part, but incurs pattern and tooling cost that must be amortised.
Fabrication
Welded construction from plate and standard section. Preferred for one-offs and very small quantities. Section changes are expensive, so fabricated parts favour simpler, more uniform geometry.
Smaller elements such as rocker arms may also be forged, which produces excellent grain flow around the load path at the cost of die tooling.
Few engineers will ever be asked to design a coupling, knuckle joint or lever from scratch. The value of doing so is that the same principles — identify the load path, proportion from a governing dimension, then check every stress on every component in that path — apply to designing any machine element for any purpose.
Rigid coupling proportions
For steel or cast iron couplings joining steel shafts, the following proportions expressed in terms of shaft diameter d in millimetres produce a sound design.
| Feature | Dimension | Proportion |
|---|---|---|
| Flange | Outside diameter | 2.6 d + 75 |
| Flange | Internal width | Nut thickness + 3 |
| Flange | Radial thickness | 5 to 10 mm |
| Hub (boss) | Length | 1.3 d + 3 |
| Hub (boss) | Diameter | 1.8 d + 5 |
| Web | Width | 0.33 d |
| Bolts | Number | 3 + 0.025 d, rounded |
| Bolts | Diameter | 0.25 d, rounded to a standard size |
| Bolts | Pitch circle diameter | 2.2 d + 35 |
Notice the structure of these expressions. The multiplier scales with torque capacity, which grows with shaft diameter; the constant provides the minimum practical size that even a tiny coupling needs for bolt heads, spanner access and casting soundness. That is why the relationships are affine rather than purely proportional.
Knuckle joint: proportions and complete stress analysis
A knuckle joint transmits tension — and sometimes compression — between two rods while permitting angular movement in one plane. It comprises a rod with an eye, a rod with a fork, and a pin through both.
The load path, component by component
| Component | Checks required | Notes |
|---|---|---|
| Rod | Tensile stress; buckling if compression also occurs. | Both ends are pinned, so there is no bending or shear in the rod. For a round rod the radius of gyration is d/4. Apply a safety factor to the critical buckling force. |
| Pin | Bending, shear, bearing. | Bending governs. Maximum shear stress in a pin in bending exceeds the average shear stress; base bearing stress on the weaker of pin and eye or fork material. |
| Eye | Tensile, shear, bearing. | Tensile across the reduced section; shear on the two areas ahead of the pin; bearing on the projected pin area. |
| Fork | Tensile, shear, bearing. | Same expressions as the eye, with 2a substituted for b. In a standard-proportioned joint of the same material, 2a exceeds b, so the fork is less critical than the eye. |
The concentrated loading assumption places equivalent point loads at the mid-points of fork and eye; the distributed assumption spreads them uniformly. The concentrated version is the more conservative and generally produces a pin larger than necessary, so the distributed expression is recommended for design. Both assume zero side clearance — where clearance exists the bending moment is greater and must be recalculated from the actual geometry.
In the basic joint there is no separate bearing, and rotation or oscillation occurs directly between pin and eye or pin and fork. Where movement is considerable, fit bearings: the pin is then made a tight fit in the eye, or secured with a grub screw, and plain or rolling element bearings are provided in the fork.
Lever design
Levers follow standard design procedure and are best learned by example, but a few principles recur in every case.
Bending governs
Bending stress is usually the critical stress, so the section is made deeper than it is wide. Where weight matters, an I section is used.
Maximum moment at the fulcrum
Bending stress peaks at the fulcrum — or, where the lever has an integral boss, just outside the boss. That is the section to check, not the geometric centre.
Fulcrum bearing pressure
Frequently the critical design factor where the lever performs many operating cycles. Rolling element bearings in some cases, plain journal bearings in others.
Lubrication when there is no bearing
Where wear is not critical the lever runs without a separate bearing. Good practice is then to fit the boss with grease nipples or oil holes so lubricant can be applied.
Levers frequently transmit force through knuckle joints or similar connections. Those are designed by calculating the key stresses as set out above, and apportioning the remaining dimensions in good proportion.
The general method
- Identify the load pathEvery component between the applied load and the reaction, in order.
- Choose the governing dimensionUsually the shaft or rod diameter, sized on the primary stress.
- Proportion the remaining geometryFrom established relationships to that governing dimension.
- Select the manufacturing methodCast, fabricated or forged — and adjust the geometry to suit it.
- Check every stress in every componentTension, shear, bearing, bending and buckling as applicable to each.
- Identify the critical componentThe one closest to its allowable stress. It governs the design factor of the whole assembly.
- Adjust and re-checkChanging one dimension changes the proportions of everything downstream.
- Confirm it looks rightIf the proportions look wrong after the analysis, the analysis usually contains an error — or the design is genuinely unusual and warrants explanation on the drawing.
Good proportions are the accumulated experience of many designs that worked. They fail where the design uses unusual materials, faces space restrictions, carries an atypical load spectrum, or operates in an environment the original experience never covered. In those cases the analysis is not a check on the proportions — it is the design.
Design checklist
- Load path identified component by component, end to end.
- Governing dimension established from the primary stress calculation.
- Remaining geometry proportioned from established relationships.
- Manufacturing method selected and geometry adapted to suit it.
- Rod checked in tension and, where compression occurs, in buckling.
- Pin checked in bending, shear and bearing, using the recommended moment expression.
- Side clearance accounted for in the pin bending moment where it exists.
- Eye and fork both checked in tension, shear and bearing.
- Bearing stresses based on the weaker of the two materials in contact.
- Levers checked at the fulcrum or just outside the boss, not at mid-span.
- Fulcrum bearing pressure assessed where the cycle count is high.
- Lubrication provision included where no separate bearing is fitted.
- Final geometry sense-checked against the appearance test.