Like most other fundamental mechanisms, metal springs have existed since the Bronze Age. Even before metals, wood was used as a flexible structural member in archery bows and military catapults. Precision springs first became a necessity during the Renaissance with the advent of accurate timepieces. The fourteenth century saw the development of precise clocks which revolutionized celestial navigation. World exploration and conquest by the European colonial powers continued to provide an impetus to the clockmakers' science and art. Firearms were another area that pushed spring development.
The eighteenth century dawn of the industrial revolution raised the need for large, accurate, and inexpensive springs. Whereas clockmakers' springs were often hand-made, now springs needed to be mass-produced from music wire and the like. Manufacturing methodologies were developed so that today springs are ubiquitous. Computer-controlled wire and sheet metal bending machines now allow custom springs to be tooled within weeks, although the throughput is not as high as that for dedicated machinery.
| Spring Constant Dependencies |
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 | | For the springs in this discussion, Hooke's Law is typically assumed to hold,
We can expand the spring constant k as a function of the material properties of the spring. Doing so and solving for the spring displacement gives,
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where G is the material shear modulus, na is the number of active coils, and D and d are defined in the drawing. The number of active coils is equal to the total number of coils nt minus the number of end coils n* that do not help carry the load,
The value for n* depends on the ends of the spring. See the following illustration for differentn* values:
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| Geometrical Factors |
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The spring index, C, can be used to express the deflection,
The useful range for C is about 4 to 12, with an optimum value of approximately 9. The wire diameter, d, should conform to a standard size if at all possible.
The active wire length La can also be used to form an expression for the deflection,
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| Shear Stress in the Spring |
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The maximum shear stress tmax in a helical spring occurs on the inner face of the spring coils and is equal to,
where W is the Wahl Correction Factor which accounts for shear stress resulting from spring curvature,
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| Risk Factors |
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Compression spring bucking refers to when the spring deforms in a non-axial direction, as shown here,
Buckling is a very dangerous condition as the spring can no longer provide the intended force. Once buckling starts, the off-axis deformation typically continues rapidly until the spring fails. As a result, it is important to design compression springs such that their likeliness to buckle is minimized.
Buckling of compression springs is similar to buckling for vertical structural columns. When the free height of the spring (Lfree) is more than 4~5 times the nominal coil diameter D, the spring can buckle under a sufficiently heavy load.
The maximum allowable spring deflection Dmax that avoids buckling depends on the free length, the coil diameter, and the spring ends (pivot ball, ground & squared, etc.).
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| Buckling Thresholds |
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One quick method for checking for buckling is to compute the deflection to free height ratio (D/Lfree) and use the following chart to check if the ratio exceeds the maximum allowable value:
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| Spring Geometry |
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Extension springs typically appear as follows,
Extension springs are typically manufactured with an initial tension Fi which presses the coils together in the free state. This fabrication method allows consistent free lengths to be produced, but since the initial tension is not zero, the spring rate is not truly linear when measured from the resting state. However once the initial tension is overcome, the spring does behave linearly.
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| Shear Stress |
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Since extension springs have an initial tension in their resting state, they also have a shear stress in their coils while at rest. The maximum shear stress (at rest) ti occurs on the inner face of the coils, and is given by the equation,
where D is the nominal diameter of the spring, d is the wire diameter, and W is the Wahl Correction Factor.
After the initial tension is overcome, the extension spring can be analyzed as a compression spring with a negative force. The maximum shear stress (tmax) in the spring increases with the load and is given by,
The spring extension D is given by,
where G is the shear modulus and nt is the total number of coils.
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| End Stress Concentrations |
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Consider a regular hook we typically see on an extension spring. The geometry of the hook often causes stress concentration which leads to failure. The following illustration shows this geometry and defines the radial parameters r1 to r4,
The maximum bending stress at point A and the maximum shear stress at point B can be expressed as follows, 
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| Extension Spring Factor of Safety |
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If a compression spring fails, catastrophic failure of the supported assembly is often prevented by the fact that the parts containing the ends of the compression spring will at worst compress the remains of the spring.
With an extension spring, there is no such safety geometry since the spring is in tension. For this and other reasons, extension spring maximum working stresses are typically limited tothree-fourths (3/4) of those for compression springs of similar geometry and material.
| First Natural Frequency Formula |
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When springs are used in a moving mechanism, their dynamic behaviors have to be analyzed. For example, valve springs in an engine have their own natural frequencies. The designers must ensure that springs are operated well under their first natural frequency at maximum engine speed, or they risk damage to the pistons because the springs may not return the valves in time.
The first natural frequency of a helical spring is found to be,
where d is the wire diameter, D is the nominal coil diameter, nt is the total number of coils, Gand r are the shear modulus and density of the spring material, respectively.
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| Derivation By Analogy |
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An easy way to derive the above equation is to draw an analogy between a rod and a spring. The analogy works because both objects are continuously distributed elements, in that their stiffness and mass are spread uniformly throughout their interiors.
Both the spring and the rod obey Hooke's Law when used in static applications, 
where DL is the change in length of the spring or rod. The stiffness k for the rod is given by,
where E is the Youngs modulus of the material, and A and L are defined in the picture above.
The change in length of the rod for dynamic applications is given by,
where the wavenumber n is given by,
and f is the driving frequency (in Hz). To check that we have the right equation, we note 2 things: The dynamic equation for DL satisfies the governing partial differential equation for the rod, and DL becomes Hooke's Law in the static limit (f goes to zero),
To find the natural frequencies for the rod, we look at where the change in length of the rod blows up. This occurs when nL in the denominator equals one of the following: {p, 2p, 3p, ...}. The first natural frequency occurs when,
We solve for fres and substitute in krod and the volume of the rod (A*L),
We recognize that the density times the volume equals the mass of the rod. We can therefore simplify the resonant frequency formula to,
By analogy, the spring's first natural frequency will have the same equation,
where k is now the spring stiffness, and M is the spring mass (which can be found by weighing the spring).
Note that this equation is similar to that for a lumped spring-mass oscillator. However, the frequency here is a factor of p higher due to the distributed mass in the spring (and rod).
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| Expressed in Terms of Spring Geometry |
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We can express the natural frequency for the spring in terms of its geometry and shear modulus (instead of its overall stiffness k and its mass).
To do so, we find the volume of the spring,
and note the stiffness of the spring in terms of its geometry and shear modulus G and number of active coils na,
Substituting these two equations into the formula for fres gives,
If the spring has several coils, we can assume that the number of active coils equals the number of total coils. We can also allow the following numerical approximation,
These two approximations give us our final equation for the spring resonant frequency,
To use this formula we need to know the material's G and all of the spring geometry. It's much easier to use the formula from the last section which only needs the spring stiffness and mass, especially when working with springs where the material is uncertain.
Article resource: http://www.efunda.com
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