Machine Design : Design of Springs, Shafts & Gears

By Shivendra Pratap|Updated : February 7th, 2022


Spring is defined as an elastic machine element, which deflects under the action of the load and returns to its original shape when the load is removed.

Basic Functions of Spring

  • Cushioning, absorbing, or controlling of energy due to shock and vibration. example: Car
  • Springs or railway buffers
  • To control energy, springs-supports and vibration dampers.
  • Measuring forces, spring balances, gauges.
  • Storing of energy, in clocks or starters, the clock has spiral type of spring which is wound to coil and then the stored energy helps gradual recoil of the spring when in operation.

Materials Used for Spring: One of the important considerations in spring design is the choice of the spring material. Some of the common spring materials are given below.

  • Hard-drawn wire: This is cold drawn, cheapest spring steel. Normally used for low stress and static load. The material is not suitable at subzero temperatures or at temperatures above 120°C.
  • Oil-tempered wire: It is a cold drawn, quenched, tempered, and general purpose spring steel. However, it is not suitable for fatigue or sudden loads, at subzero temperatures and at temperatures above 180°C. When we go for highly stressed conditions then alloy steels are useful.
  • Chrome Vanadium: This alloy spring steel is used for high stress conditions and at high temperature up to 220°C. It is good for fatigue resistance and long endurance for shock and impact loads.
  • Chrome Silicon: This material can be used for highly stressed springs. It offers excellent service for long life, shock loading and for temperature up to 250°C.
  • Music wire: This spring material is most widely used for small springs. It is the toughest and has highest tensile strength and can withstand repeated loading at high stresses. However, it cannot be used at subzero temperatures or at temperatures above 120°C. Normally when we talk about springs we will find that the music wire is a common choice for springs.
  • Stainless steel: Widely used alloy spring materials.
  • Phosphor Bronze / Spring Brass: It has good corrosion resistance and electrical conductivity. That’s the reason it is commonly used for contacts in electrical switches. Spring brass can be used at subzero temperatures.

Helical spring:

Springs are classified according to their shape.

  • The spring can be a helical coil of a wire, a piece of stamping or a flat wound-up strip
  • The most popular type of spring is helical spring
  • The helical spring is made from a wire, usually of usually of circular cross-section, which is bent in the form of a helix
  • There are two types of helical springs – compression spring and extension spring.


Fig. (a) Helical Compression spring, (b) Helical Extension spring



Fig.: Anatomy of helical spring

d = wire diameter of spring (mm),

D= inside diameter of spring coil (mm),

Do=Outside diameter of spring coil (mm),

D=mean coil diameter (mm).



Spring index (C) = Ratio of mean coil diameter to wire diameter,


Axial length of the spring which is so compressed that the adjacent coils touch each other. The spring is completely compressed and no further compression is possible.

Solid Length = Nd

where, N = total number of coils,


Axial length of the spring, which is subjected to maximum compressive force.

When the spring is subjected to maximum force, there should be some gap or clearance between the adjacent coils.

The gap is essential to prevent clashing of the coil.

Total gap = (N – 1) x gap between adjacent coils.


Axial length of an unloaded helical compression spring. In this case, no external force acts on the spring.


Force required to produce unit deflection,


where, P = axial spring force (N)

δ= axial deflection of the spring corresponding to the force  (mm)

Spring Index

  • Spring index is defined as the ratio of the mean coil diameter to the wire diameter of the spring and denoted as “C”.

C = D/d = Mean coil diameter / Wire diameter

 Shear stress correction factor  is defined as,


Wahl correction factor  is defined as,


Resultant shear stress in the spring wire,


Axial deflection ‘ δ ’ of the spring:


Stiffness of spring:



Design of Shaft:

  • A shaft is a machine component used to transmit rotary motion or torque. Shafts transmit torque using the gears, belts, pulleys etc.
  • Shafts are generally subjected to bending moment, torsion ,and axial force or a combination of these three.

    Axle: A stationary member used as support for rotating elements such as wheels, idler gears, etc.

    Spindle: A short shaft or axle (e.g., the headstock spindle of a lathe).

     Line shaft: A shaft having connection with a prime mover and transmitting power, to one or many machines, is called the line shaft.

    Jackshaft: A short shaft connecting a prime mover with a line shaft, is called the jackshaft.

    Flexible shaft: It is a connection used for power transmission between two members having shafts axes at an angle with each other.


    • Most shafts are round but they can come in many different shapes including square and octagonal. Keys and notches can also result in some unique shapes.

    Hollow Shafts Vs Solid Shafts

    • Hollow shafts are lighter than solid shafts of comparable strength but are more expensive to manufacture.
    • Thus, hollow shafts are used primarily for applications where weight is critical. Example: rear-wheel drive cars propeller shafts should be lightweight to handle speeds within the operating range of the vehicle.

    Shaft Design: Shafts are designed based on either the strength or rigidity or both.

    Design based on Strength: The design in this method ensures that stresses in the shaft does not exceed the material yield stress.

    Based on Rigidity: It is based on the principle that shaft deflection due to bending and maximum twist (due to torsion) is within the permissible limits.

    Shaft Equations: The following equations given below are general equations and  modifying factors such as loading factors, pulsating power source factors, safety factors, and stress concentration factors can be used as per application.

    Torsional shear stresses: A machine component acted upon with two equal and opposite couples acting in parallel planes is called under the torsion. The induced internal stresses to resist the twist, are called torsional shear stresses.


     The torsion equation is given as :


     Basic equations in torsion:

  • Solid round shaft: byjusexamprep

    Hollow round shaft:byjusexamprep

    Strength criteria:

    The strength criteria uses the first two terms of torsion equation and design is done on basis that stress induced in the shaft must not exceed the strength of material of shaft.


    Rigidity criteria: The rigidity criteria uses the last two terms of torsion equation and design is done on the basis that maximum angular twist must exceed a certain value.


     Bending stresses in shafts:

    A beam or a member is said to be under pure bending when it is subjected to two equal and opposite couples in a plane along the longitudinal axis of the beam (i.e. bending couples) in such a way that magnitude of bending moment remains constant throughout the length of the beam.




    Chances of failure decrease.

    • Cross-section which has higher section modulus is best suitable under bending [Bending Stress should be minimum]


    Gears are defined as toothed wheels which transmit power and motion from one shaft to another by means of successive engagement of teeth.


    Gears are broadly classified into three groups, viz.,

    • Spur Gears
    • Helical Gears
    • Bevel Gears

    The torque transmitted by the gears is given by, 



    P = Power transmitted by gears (kW)

    N=speed of rotation (rpm),

    The tangential component (Pt), 


    The radial component,


    The resultant force PN  is given by,



    Lewis equation is considered as the basic equation in the design of gears.


    Fig.: Gear tooth as cantilever beam

    The Lewis equation is based on the following assumptions:

    • The effect of the radial component, which induces compressive stresses, is neglected
    • It is assumed that the tangential component is uniformly distributed over the face width of the gear
    • The effect of stress concentration is neglected
    • It is assumed that at any time, only one pair of teeth is in contact and takes the total load

     The cross-section of the tooth varies from the free end to the fixed end. Therefore, a parabola is constructed within the tooth profile.



    The equation is written as,


     Y is called the Lewis form factor,


    When the stress reaches the permissible magnitude of bending stresses, the corresponding force Pt is called the beam strength.

    Therefore, the beam strength (Sb) is the maximum value of the tangential force that the tooth can transmit without bending failure.


    This equation is known as Lewis equation.

     It is observed that m and b are same for pinion as well as for gear.

    • When the different materials are used, the product of decides the weaker between the pinion and gear.
    • The Lewis form factor is always less for a pinion compared with gear.
    • When the same material is used for the pinion and gear, the pinion is always weaker than the gear.

     The service factor Cs is defined as,


    The effective load between two meshing teeth is given by,


    In order to avoid failure of gear tooth due to bending,

    Sb > Peff


    In order to avoid this type of failure, the proportions of gear tooth and surface properties, such as surface hardness should be selected in such a way that the wear strength of the gear tooth is more than the effective load between the meshing teeth.



     Q ratio factor is defined as,


    K is a load-stress factor is defined as,


    d'p=diameter of pinion,

    b= Width of gear,

    When the tangential force is increased, the contact stress also increases.

    Pitting occurs when the contact stress reaches the magnitude of the surface endurance strength. The corresponding value of Pt is called wear strength. Therefore, the wear strength is the maximum value of the tangential force that the tooth can transmit without pitting failure.

    Replacing Pt by Sw,

  • byjusexamprep

    Sw=wear strength of the gear tooth(N).

  • The expression for the load-stress factor K can be simplified when both the gears are made of steel with a 20º pressure angle,


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