Springs
Springs are elastic members that exert forces, or torques, and absorb energy, which is usually stored and later released. The force produced by a spring can be compressive or tensile and linear or radial.
Springs are usually, but not necessarily, made of metal. Plastics can be used when loads are light. Modern structural composites are being introduced for some applications requiring minimum spring mass. Blocks of rubber often constitute springs, as in bumpers and in vibration isolation mountings of various machines such as electric motors and internal combustion engines. Pneumatic springs of various types take advantage of the elastic compressibility of gases, as compressed air in automotive "air shocks" and as hermetic ally sealed highpressure nitrogen gas in the hydropneumatic suspensions of French Citroen automobiles. For applications requiring compact springs providing very large forces with small deflections, hydraulic springs have proved effective. These work on the basis of the slight compressibility of liquids, as indicated by their bulk modulus of elasticity. Product cost can sometimes be reduced by designing the required elasticity into other parts, rather than making those parts rigid and adding a separate spring.
Springs can be classified by the direction and nature of the force exerted when they are deflected. Several types of spring are listed in Table 1 according to the nature of force or torque exerted. The principal characteristics of various classes of springs are summarized in Table 2.
Table 1 Classification of springs according to the nature of force or torque exerted
Actuation 
Type of spring 
Compressive 
Helical compression springs 
Tensile 
Helical extension springs 
Radial 
Garter springs 
Torque 
Torsion springs 
Table 2 Principle characteristics of a variety of types of spring
Type of spring 
Principle characteristics 
Helical compression springs 
These are usually made from round wire wrapped into a straight cylindrical form with a constant pitch between adjacent coils 
Helical extension springs 
These are usually made from round wire wrapped into a straight cylindrical form but with the coils closely spaced in the no load condition. As an axial load is applied the spring will extend but resisting the motion 
Drawbar springs 
A helical spring is incorporate into an assembly with two loops of wire. As a load is applied the spring is compressed in the assembly resisting the motion 
Torsion springs 
These exert a torque as the spring id deflected by rotation about their axis. A common example of the application of a torsion spring is the clothes peg 
Leaf springs 
Leaf springs are made from flat strips of material and loaded as cantilever beams. They can produce a tensile or compressive force depending on the mode of loading applied 
Belleville springs 
These comprise shallow conical discs with a central hole 
Garter spring 
These consist of coiled wire formed into a continuous ring so that they can exert radial inward force when stretched 
Volute springs 
These consist of a spiral wound strip, that functions in compression. They are subject to significant friction and hysteresis 
Spring design involves the relationship between force, torque, deflection, and stress. Springs have many uses in connection with machine design, such as to cushion impact and shock loading, to store energy, to maintain contact between machine members, for force measuring devices, to control vibration, and other related functions.
Helical springs
Figure 1(a) shows a helical coil spring with ends adapted to support a compressive load. At (b) the spring has been deflected by the axial load P, which may be assumed to compress the spring as between two parallel plates. The notation applies:
P = axial load, N D = mean diameter of coil, mm = (D_{o} + D_{i}) / 2 d = diameter of wire, mm = (D_{o} – D_{i}) / 2 p = pitch of coils, mm δ = deflection of spring, mm n = number of active coils C = spring index = D/d = (D_{o}d)/d = (4 : 12); less that 4 it is difficult to manufacture, more than 12 is likely to buckle. G = torsional modulus of elasticity, N/mm^{2} τ_{s}_{ }= shearing stress, N/mm^{2} 

The number of active coils, n, represents the coils in the spring with the exception of those which lie flat against the compression plats. The inactive coils do not contribute to the deflection of the spring.
Springs (DIN ISO 2167)
Representation of a cylindrical helical compression spring made out of round wire. A in view B in section and C as name plate drawing

Stress in a helical spring made of round wire
Figure 2 shows a part of a compression spring that supports a compressive spring that supports a compressive load P and a section of the wire cut by axial plane. The part of spring shown in the figure is in equilibrium under the action of the two forces P and resisting torsional moment T. The torque equals
_{}
The shearing stress due to the torque T is
_{}
and the direct shearing stress is
_{}
where
_{} and _{}
for solid circle cross section
The maximum shearing stress, which is located at the inner side of the curved wire, may determined by
_{}
where
_{} is the direct shear factor
It is evident form the above equation that for springs of small index C the effect of direct shear (1/2C) is appreciable.
The effect of curvature of the wire as it forms the coil should be considered also.
In order to include the effects of both direct shear and wire curvature, a stress factor had been determined by the use of approximate analytical methods developed by A. M. Wahl which may be used in the above equation to determine the maximum shearing stress in the wire as follows:
_{}
_{}
where
_{}
Deflection of helical spring
An equation for the axial deflection of a helical spring in terms of the axial load, spring dimensions, and a materials constant may be conveniently determined by equating the work re3fquired to deflect the spring to the strain energy in the twisted wire. For closecoiled springs the bending of the wire is small and the strain energy of bending may be neglected.
The axial load as shown in Fig. 2 increases linearly from zero to P, and hence the work required to compress the spring is the average force P/2 times the deflection, or Pδ/2.
The strain energy in a bar twisted by a torsional moment T through a total angle θ is Tθ/2. The total angle of twist θ = TL/GJ, where L equals the length of the twisted wire and J equals the polar moment of inertia of the wire section. The active length of the wire in the helical spring equals (πDn/cos α), where α is the lead angle of the helix, which for closecoiled spring is of the order of 5^{o}; hence cos α is unity.
For springs of indexes as used in machinery, namely, 2.5 or 3 and over, the strain energy in the curved wire forming the coils will be closed to the strain energy in a straight wire; hence from the above relations the following equation for δ may be obtain
_{}
where
T = P D/2
L = π D n
J = π d^{4}/32
_{}
From the above equation
_{}
where (P/δ) is known as the spring rate. The spring rate represents the slope of the loaddeflection line as shown in Fig. 3. Any difference between the ideal line and the actual curve is due principally to the change in number of active coils as the spring is deflected, to mechanical hysteresis of the material, and the friction between the end coils in contact with each other or with the loading plate.
Compression springs in which the free length is more than four times the mean diameter of the coils may fail by sidewise buckling.
The spring rate for springs in parallel:
For springs having rates as shown in figure (a)
_{}
The spring rate for springs in series:
For springs having rates as shown in figure (b)
_{}
Nomogram for helical spring calculation: (G = 81,400 MPa)
Explanation of the nomogram:
The Nomogram applies to cylindrical helical extension and compression springs made of round steel wire (shear modulus G=81,400 N/mm^{2}). In the case of materials with a different shear modulus G^{’} spring deflection must be multiplied by G/G^{’}.
The nomogram indicates the deflection s^{’} of one coil. Total deflection s is obtained by multiplying (s^{’}) by the number of active coils n: s = n . s^{’}.
Spring ends:
Helical spring ends may be either plain, plain ground, squared, or squared and ground as shown in Fig. 4 below. This results in a decrease of the number of active coils and affects the free length and solid length of the spring as shown below.
p=(D/3 : D/4), n= ( 3: 15)
Buckling:
Buckling may occur in compression springs if the free length is over 4 times the mean diameter unless the spring is properly guided. The critical axial load that will cause buckling may be approximated by
F_{cr} = k L_{f} K_{L}
where:
F_{cr} = axial load to produce buckling, N
k = spring rate, N/m, of axial deflection
L_{f} = free length of the spring, m
k_{L} = a factor depending on the ratio L_{f}/D (slenderness factor).
or L_{f}/D < 3 for no buckling
Buckling depends on
L_{f}/D and δ/L_{f}
Buckling can be avoided by providing a spring guide

Allowable stresses:
For helical chromevanadium steel springs, oiltempered, hot wound and heattreated after forming, are tabulated below.
Sever service includes rapid continuous loading where the ratio of minimum to maximum stress is one half or less. Average service is the same as severe except for intermittent operation. Light service includes springs subjected to static loads or to infrequently varied loads.
Spring design approach
· Specific material
· Guess a trail diameter (space available)
· Check spring rate and free length
 spring rate: k = (F_{o}F_{i}) / (L_{i}L_{o})
 spring free length L_{f} = L_{i }+ (F_{i}/k)
where:
F_{o} is the operating force, and L_{o }is the operating length
F_{i} is the installed force, and L_{i} is the installed length
Leaf Springs
Acts as:
 an elastic element
 guide
 damper
+ they are easy to manufacturer and convincement to repair
Disadvantage:
 large metal content (energy stored in unit volume in spring or torsion bar is 4 times more than a leaf spring; that lead to increase metal content).
 sufficient unsprung mass
 short service life.
If it is desired to maintain uniform bending stresses over the length of the beam then the width of the cantilever needs to vary linearly with the location as illustrated in Figure i, (σ_{b} = M y/I = 12 F L y / b h^{3}). The concept used in producing compact cantilever springs of uniform bending stress is to chop the triangular form illustrated in Figure i, into a number of strips and recombine them as illustrated in Figure ii. The multileaf spring shown and the single triangular section beam both have the same stress and deflection characteristics with the exceptions that the multileaf spring is subject to additional damping due to friction between the leaves and that the multileaf spring can carry a full load in only one direction due to the tendency for the leaves to separate. Leaf separation can be partially overcome by the provision of clips around the leaves.


The deflection of a triangular leaf spring is given by
_{}
where
F is force (N)
L is the length (m)
E is the Young’s modulus (N/m^{2})
I is the second moment of area (m^{4})
For a rectangular crosssection,
_{}
where b is width (m); and h is thickness (m).
The spring rate is given by
_{}
The corresponding bending stress (for cantilever, semielliptic and fullelliptic) is given by
_{}
For semielliptic beam the maximum deflection at the center is given by
_{}
For a fullelliptic beam the maximum deflection at the center is given by
_{}
* in the above equation the length b can be changed by nb^{’} where n is the number of leafs and b^{’} is the width of single leaf. The leaf width b^{’} is chosen from the available range of rolled products. It is desired that the following condition should hold
6 < b^{’}/h < 10
6 < n < 14
The spring length can be determined using the equations for spring stiffness and stress to obtain
_{}
where
δ = is the total deflection (δ_{st} + δ_{d})
 Effect of adding an extra full length number of leaves (n_{e}) to the graduate length number of leaves (n_{g}):
_{}
That means the extra leaves will have more stress than graduate ones.
_{}
Torsion beam suspension (torsion bars)
Definition: A steel bar that is twisted to support the weight of the vehicle. Torsion bars are used in place of coil or leaf springs on some vehicles, and allow ride height to be adjusted to compensate for sage that occurs over time.


The bars are usually solid of circular cross section although hollow tubes and rectangular bars are used.
_{}
τ_{all} = 1000 1050 MPa
The angle of twist and stiffness of a torsion bar are expressed as
_{}
The torsionbar working length L (without spline ends) is determined by the θ equation. It is recommended to choose the diameters and lengths of spline ends depending on the torsionbar diameter
d_{sp} = (1.2 1.3)d; and L_{sp}= (0.61.2)d
Suspension Formulae
_{}
where b is the width of spring blade (m), L is the distance between the eyes of the spring when laden (m), t is the thickness of the blade (m), n is the number of blades, and E is the modulus of elasticity, which (modified to allow for internal friction) is 159 x 10^{6} kN/m^{2}.
For a torsion bar, the spring rate is given as the twisting moment per angular deflection. When a lever is added, this can be converted into a rate for the vertical deflection of the end of the lever.
Spring rate (torsion bar, for deflection at end of lever)
_{}
where G is the modulus of rigidity, which is 78.5 x 10^{6} kN/m^{2} in this case, d is the diameter of the torsion bar (m), l is the effective length of the torsion bar (m), i.e. half the length of the bar for an antiroll bar, and e is the length of the lever (m).
Spring rate (coil spring)
_{}
where G is the modulus of rigidity, which is 81.5 x 10^{6} kN/m^{2} in this case, d is the wire diameter (m), n is the number of free coils, and D is the mean coil diameter (m). To find the number of free coils it is necessary to subtract the number of dead coils form the total number of coils. The dead coils are those that provide the abutment and so cannot be deflected, usually 1.5 to 2 coils.
Type 
Configuration 
Action 