Part VIII: Design of Machine Elements

Introduction to Machine Design

Machine design is the process of creating engineering components and systems capable of performing their intended function safely, economically, and reliably. A machine element must be designed to withstand the static and dynamic loads it experiences during service, with an appropriate factor of safety to account for uncertainties in loading, material properties, and manufacturing.

The factor of safety (FOS) is defined as the ratio of the material's ultimate (or yield) strength to the design (working) stress. A higher factor of safety is used where loads are uncertain, where failure consequences are severe, or where the material's properties are not well characterised; a lower factor of safety may be used for well-understood, static, ductile applications. Typical factors of safety range from about 1.5–2 for steady loads on ductile materials in well-controlled conditions, up to 5–10 or more for shock loads, brittle materials, or safety-critical applications.

Types of Loads and Stresses

Machine elements experience various types of loading: static loads (constant, unchanging with time), which are the simplest to design for; and dynamic (fluctuating) loads, which include repeated, reversed, and fluctuating stresses that can cause fatigue failure even at stress levels well below the material's static strength.

Fatigue is the progressive, localised structural damage that occurs when a material is subjected to cyclic loading. Fatigue failure typically initiates at a stress concentration point (a notch, keyway, fillet, or surface defect), propagates as a crack under repeated loading, and finally results in sudden fracture once the remaining cross-section can no longer bear the load. The S-N curve (stress amplitude versus number of cycles to failure) is the standard tool for characterising a material's fatigue behaviour; for many ferrous materials, the S-N curve flattens out at a certain stress level called the endurance limit, below which the material can theoretically withstand an infinite number of cycles without fatigue failure.

Stress concentration occurs at any geometric discontinuity — a hole, keyway, fillet, groove, or sudden change in cross-section — where local stress rises well above the nominal (average) stress calculated from simple load/area formulas. The stress concentration factor (Kt) is the ratio of maximum local stress to nominal stress. Stress concentration effects are especially critical under fatigue loading, since fatigue cracks almost always initiate at points of stress concentration. Designers reduce stress concentration by using generous fillet radii, avoiding sharp corners, and using stress-relieving grooves where sudden section changes are unavoidable.

Shafts, Keys, and Couplings

A shaft is a rotating machine element used to transmit power from one part of a machine to another, typically subjected to combined bending and torsional loading, sometimes with axial loading superimposed. Shaft design uses theories of failure such as the maximum shear stress theory (for ductile materials under combined bending and torsion) to determine the required diameter for a given combination of bending moment, torque, and allowable stress. An axle, by contrast, is a non-rotating (or rotating but non-power-transmitting) member that primarily supports transverse loads, such as a railway wagon axle, and is designed mainly for bending rather than torsion.

A key is a machine element used to connect a rotating machine element (such as a gear or pulley) to a shaft, transmitting torque between them while preventing relative rotation. Common key types include the sunk key (rectangular or square, fitted in matching keyways in both the shaft and hub), the saddle key (fitted only in the hub keyway, resting on a flat on the shaft, used for light loads), and the woodruff key (a semicircular key, self-aligning, commonly used in machine tool and automotive applications).

A coupling connects two shafts together so that power can be transmitted from one to the other, while accommodating some degree of misalignment. A rigid coupling (such as a flange coupling) connects shafts that are precisely aligned and does not tolerate misalignment. A flexible coupling (such as a bushed-pin flange coupling or an Oldham coupling) can accommodate a small amount of angular, parallel, or axial misalignment between the connected shafts, and also helps absorb shock and vibration.

Springs

A spring is a resilient machine element designed to store and release mechanical energy through elastic deformation. Helical compression springs and helical tension (extension) springs are the most common types, made by coiling wire around a mandrel; they store energy in torsion within the wire cross-section as the coil is compressed or extended. Leaf springs, commonly used in vehicle suspensions, consist of a stack of flat plates ("leaves") of varying length clamped together, storing energy through bending. Torsion springs store energy through twisting about their axis and are commonly used in applications like clothes pegs, mousetraps, and door hinges.

The spring stiffness (rate), k, is the load required to produce unit deflection, and is a key design parameter for helical springs, dependent on wire diameter, coil diameter, number of active coils, and the shear modulus of the spring material. Spring index is the ratio of mean coil diameter to wire diameter; a low spring index (tightly coiled spring) results in higher stress concentration at the inner surface of the coil, requiring a stress correction factor (such as the Wahl correction factor) in accurate design calculations.

Bearings

A bearing is a machine element that supports a rotating or reciprocating shaft, allowing relative motion while carrying the load transmitted between the shaft and its housing, and minimising friction. Bearings are broadly classified as sliding contact (plain/journal) bearings, in which the shaft rotates within a bore with a thin film of lubricant separating the surfaces, and rolling contact (antifriction) bearings, in which rolling elements (balls or rollers) are interposed between the inner and outer races, substantially reducing friction compared to sliding contact.

Rolling contact bearings are further classified by the type of rolling element and the direction of load they primarily carry: deep groove ball bearings (versatile, carry combined radial and moderate axial load), angular contact ball bearings (designed for combined radial and substantial axial load in one direction), cylindrical roller bearings (high radial load capacity, minimal axial capacity), tapered roller bearings (carry combined heavy radial and axial loads, commonly used in vehicle wheel hubs), and thrust bearings (designed to carry axial load exclusively, with the rolling elements arranged perpendicular to the shaft axis).

Bearing life is commonly expressed using the L10 life, the number of revolutions (or hours, at a given speed) that 90% of a group of apparently identical bearings will complete or exceed before the first evidence of fatigue failure appears. The L10 life is inversely related to the cube (for ball bearings) or a similar power (for roller bearings) of the applied load, meaning even a modest reduction in load significantly extends bearing life.

Gears — Design Considerations

Beyond the basic classification and terminology covered in the Theory of Machines section, gear design requires attention to the module (the ratio of pitch circle diameter to number of teeth, a standard measure of tooth size), the pressure angle (commonly 20° in modern involute gear systems, replacing the older 14.5° standard), and the law of gearing, which requires that the common normal to the tooth profiles at their point of contact must always pass through the pitch point, ensuring a constant velocity ratio throughout mesh.

Interference in gear teeth occurs when the tip of one gear's tooth digs into the root fillet of the mating gear's tooth, which can occur when the number of teeth on the pinion is too small relative to the gear. Interference can be avoided by using a minimum number of teeth on the pinion (dependent on the pressure angle), by increasing the pressure angle, or by using profile modification techniques such as tooth undercutting or addendum modification.

Practice Questions — Design of Machine Elements

1. The factor of safety is defined as the ratio of a material's strength to its:
   (a) Design (working) stress (b) Modulus of elasticity (c) Poisson's ratio (d) Density
2. Fatigue failure in machine components typically initiates at:
   (a) A point of stress concentration (b) The exact geometric centre of the component (c) A point of zero stress (d) The component's centre of mass only
3. The endurance limit of a material is the stress level below which the material can theoretically withstand:
   (a) An infinite number of load cycles without fatigue failure (b) Only one load cycle (c) Exactly 100 load cycles (d) No load cycles whatsoever
4. A shaft, unlike an axle, is primarily designed to transmit:
   (a) Torque (along with bending) (b) Only bending moment, with no torque at all (c) Only axial thrust (d) No load of any kind
5. A woodruff key is characterised by its:
   (a) Semicircular shape, providing self-alignment (b) Rectangular cross-section identical to a sunk key (c) Complete absence of any defined shape (d) Function only as a decorative shaft feature
6. A rigid coupling is best suited for shafts that are:
   (a) Precisely aligned, with no misalignment to accommodate (b) Significantly misaligned, requiring flexibility (c) Never rotating at all (d) Made of dissimilar materials only
7. A leaf spring stores energy primarily through:
   (a) Bending of its stacked leaves (b) Torsion of a coiled wire (c) Pure axial compression only (d) Pure shear only, with no bending
8. The spring index of a helical spring is the ratio of:
   (a) Mean coil diameter to wire diameter (b) Number of coils to wire diameter (c) Free length to solid length (d) Spring stiffness to wire diameter
9. Rolling contact bearings, compared to sliding contact bearings, generally offer:
   (a) Lower friction, especially at starting (b) Higher friction under all conditions (c) No difference in friction whatsoever (d) Function only in axial loading, never radial
10. Tapered roller bearings are particularly suited to carry:
    (a) Combined heavy radial and axial loads (b) Only pure axial load, with no radial capacity (c) Only pure radial load, with no axial capacity (d) No load at all, only guiding function
11. The L10 bearing life represents the life that:
    (a) 90% of a group of bearings will complete or exceed before fatigue failure (b) 100% of bearings will exceed with absolute certainty (c) Exactly 10% of bearings will exceed (d) Is entirely independent of applied load
12. Gear tooth interference can be reduced by:
    (a) Increasing the pressure angle or using profile modification (b) Decreasing the pressure angle to zero (c) Reducing the module to zero (d) Removing all teeth from one gear entirely

Answer Key: 1.(a) FOS = strength/working stress. 2.(a) Fatigue cracks initiate at stress concentration points. 3.(a) Below the endurance limit, theoretically infinite cycles are survivable. 4.(a) A shaft transmits torque along with bending; an axle mainly carries bending. 5.(a) A woodruff key is semicircular and self-aligning. 6.(a) Rigid couplings suit precisely aligned shafts. 7.(a) Leaf springs store energy through bending of stacked leaves. 8.(a) Spring index = mean coil diameter / wire diameter. 9.(a) Rolling contact bearings have lower friction, especially at starting. 10.(a) Tapered roller bearings carry combined heavy radial and axial loads. 11.(a) L10 life is the life 90% of bearings equal or exceed. 12.(a) Increasing pressure angle or profile modification reduces interference.