Cylindrical springs of a stretching and compression

Design procedure of springs
(settlement formulas, an example of calculation of a spring of a tension, the table of shear moduli of materials, systems номограм for fast (alternative) estimations of parametres of a spring)

System of nomograms for design calculation of a twisted spring

Parameters

Red color on illustration allocates determined parameters of a spring

The following parameters of a spring are determined:

L - change of length of a spring under loading (spring stroke), mm;

d - diameter of a wire, mm;

Dm - average coiling diameter of wire, mm;

С - index of a spring;

G - shear modulus, kilogram-force/mm² (kgf/mm²) (1 kgf/mm² = 9.81 MPa);

i - number of coils;

Fmax-Fmin - difference between the maximal (matches to the maximum moving of a spring in the mechanism) and minimal (the length of a spring begins to change) loadings on a spring, kilogram-force (kgf), (1 kgf = 9,81 N);

Fmin1 - the minimal effort (40% from maximal Fmax1), kilogram-force (kgf);

Fmax1 - the maximal effort, kilogram-force (kgf);

Fmin2 - the minimal effort (60% from maximal Fmax2), kilogram-force (kgf);

Fmax2 - the appropriate maximal effort, kilogram-force (kgf).

Calculation procedure

Design formuls

The first formula:

L = m · i · (Fmax - Fmin),

where   L - elastic moving of a spring (spring stroke), mm;

            m - spring compliance (one coil), mm/kgf;

             i  - number of working coils;

            Fmin  - bootstrop (initial loading), kgf;

            Fmax - the maximal loading, kgf.

For tensioning springs, the initial load can be the preload of the spring during winding, Fmin = F₀.

Spring compliance (one coil)

m = 8 · C³ / (G · d),

where G - shear modulus, kgf/mm²;

           d - diameter of a wire, mm;

           С - index of a spring

С=Dm / d.

✍ On a note! The thinner the wire, the more flexible the coil springs can be. The more flexible the spring, the higher the spring index and the number of turns.

Recommended values of an index of a spring:

For springs of square and right-angled cross-section condition performance is recommended   D/b ≥ 4,   where b is the side of the section perpendicular to the axis of the spring.

Shear modulus  G for steel springs - (7.7 ... 8.5)*10³ kgf/mm² (1 kgf/mm² = 9.81 MPa).

The minimal loading choosing on purpose of the spring

Fmin = (0.3 ... 0.8) Fmax.

Diagrams are constructed for two variants: F1min = 40% F1max  and  F2min = 60% F2max.

Value of a maximum load (limiting)

Fext = (1.05 ... 1.2) Fmax

Relative inertial backlash of a spring of compression

j=1-Fext/Fmax.

For springs of compression I and II classes                j = 0.10 ... 0.25.

For springs of compression III class                           j = 0.15 ... 0.40.

For springs of a stretching                                           j = 0.05 ... 0.10.

So, using the systems of nomograms (or calculation), the values of forces, stroke, index, number of spring turns, etc. were found. Let's calculate the axial dimensions of the spring (nomograms for their determination are not shown).

Formulas for calculating axial dimensions of a spring.

Spring limit deformation

Sext = Fext / b,

where b is the spring rate (the reciprocal of the compliance), kgf/mm (1 kgf/mm = 9.81 kN/m)

b = (Fmax - Fmin) / L.

Coiled spring length at maximum (limit) deformation

Lext = (i₁ + 1 - i₃) · d,

where i₁ - total number of spring turns

i₁ = i + i₂,

where i₂ - number of support turns (for tensioning spring i₁ = i);

           i₃ - number of ground coils of the compression spring.

So, the length of the compression spring in the free state

L₀ = Lext + Sext.

Length of a cylindrical spring in the non-loaded condition:

X₀=L₀+2*Hh,

where   Hh - height of one hook - (0.5 ...1) D.

After the application of maximal loading length of a spring of a stretching:

X=X₀+L.

For springs of compression to calculation number of coils add on 0.75...1 coil for each end -  i₀ = i + (1.5...2).

Length of a spring of compression at contact of coils with the account grinding each end of a spring on 0.25d

X₀=( i₀ - 0.5 ) · d.

Spring pre-deflection

Smin = Fmin / b.

Spring length at pre-deflection

Lmin = L₀ - Smin.

For tensioning springs with trailers (hooks)

Lmin = L₀ + S_min.

After applying the maximum load, the length of the tensioning spring is determined by simple summation

Lmax = Lmin + L,

where L - a spring stroke, mm.

Compression spring length after applying maximum load

Lmax = Lmin - L.

Figure - Three states of tension spring (Autodesk Inventor) 

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 System of nomograms for calculation on durability of a cylindrical spring

Parameters

Red color on illustration allocates determined parameters of a spring

d - diameter of a wire, mm;

Dm - average coiling diameter of wire, mm;

С - index of a spring;

t - maximum (limit) permissible torsional stress ([τ]_t), kgf/mm² (1 kgf/mm² = 9.81 MPa);

F - the maximal loading on a spring, kgf (1 kgf = 9.81 N). 

Calculation procedure

Formulas for calculating

The maximum torsional stress (kgf/mm²):

t_max = 8 · k · F · Dm / (3.14 · d³) < [τ]_t ,

where

[τ]_t - limit permissible torsional stress, kgf/mm²;

F - loading, kgf;

k - coefficient, (curvature of a coils)

k = 1 + 1.45/С.

✍ On a note! At pulsing loading with a small number of cycles of value [τ]_t lower in 1.25...1.5 times.

After transformations, we got the formula:

d = 1.6 · [ k · Fext · C / [τ]_t ]^0.5,

where 

Fext - as much as possible admissible (limiting) loading on a spring (on a nomograph - F), matching to a twisting ultimate stress, kgf;

C - index of a spring.

1 - tungsten wire; 2 - chrome vanadium wire; 3 - oil-hardened carbonaceous wire;

4 - сarbonaceous cold-drawn wire; 5 - monel metal wire; 6 - wire from phosphor bronze; 7 - special brass wire

Permissible stresses of twisting of materials of coil springs (X - d, mm; Y - [τ]_t , MPa)

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Systems of nomograms for calculation of weight of a cylindrical spring

Parameters

Red color on illustration allocates determined parameters of a spring

d  - diameter of a wire, mm;

Dm - average coiling diameter of wire, mm;

i   - number of working coils;

m - weight of a spring (without of a hooks), g

Calculation procedure

Formulas for calculating

Weight of a spring (approximate value) without hooks

m = 19.25 · 10^-6 · i · D_m · d²

Figure - Compression spring (Autodesk Inventor) 

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 Reference Information

Material. The main materials for the springs are high-carbon steels, steels alloyed with silicon, manganese, chromium, vanadium, nickel. To work in a chemically active environment, springs from non-ferrous alloys, beryllium bronzes, silicon-manganese bronzes are used. Springs are made of wire, tape, strip steel. Round wire is usually used, since rounds work better for torsion.

Springs, depending on the material of the wire, have different strengths. For example, springs made from chromium alloy steel are stronger than those made from steel alloyed with silicon, and the latter are stronger than those made from steel alloyed with manganese.  

At the final stage of manufacturing of springs it is used spring hardening by aging in compressed condition. In need of increase of cyclic durability the spring is strengthened by fraction.

✍ On a note! At hardening in fraction in working drawings should be specified: «To harden in fraction on technique of manufacturer».

Defects are cleaned before the spring hardening by aging in compressed condition operation.

All springs with anticorrosive electrolytic coatings (chromium, nickel, cadmium, zinc, etc.) must be exposed to spring hardening by aging in compressed condition for at least 24 hours, regardless of whether spring were hardening by aging in compressed condition prior to coating or not. For non-critical springs of all classes and springs made of silicon-manganese bronze, the duration of spring hardening by aging in compressed condition is specified in the tables of the relevant standards.

Shot blasting the springs can greatly increase their endurance. As a result of shot blasting of the surface layers, the endurance limit of the springs increases by almost 50%.

According to the operating conditions, the springs are divided into classes. Springs are also classified into three force accuracy groups. Springs of different groups are designed to work with different types of loads. Accuracy groups of springs correspond to accuracy groups for geometric parameters.

✍ On a note! The requirements for the surface quality of the springs are high, since coil fatigue is (mainly) the result of the growth of submicroscopic cracks and scratches on the surface.

Composite springs.

  In some cases (large, repeated loads with a high repetition rate), in conditions of elevated ambient temperatures, which increases the likelihood of fatigue in the material, compound springs are used, consisting of several conventional compression springs. For better mutual centering, the springs are wound with right and left windings. With a concentric arrangement of the springs, the load is distributed between the springs in proportion to their stiffness. The total load is equal to the sum of the component loads. When designing compound springs, the condition of equality of the stroke, the maximum stress of the springs and the simultaneous achievement of the ultimate compression (until the coils touch) must be met. Therefore, the indexes of the springs must be the same.

Manufacturing of springs.

  It is customary to produce springs from wire with a diameter of up to 8 ... 10 mm by cold winding. Large cross-section springs are usually hot wound.
Compression springs are wound by open winding with a gap between the turns of 10 ... 20% more than the calculated axial elastic displacements of each turn at maximum working loads. The tension springs are wound so that the initial tension between the turns is provided. The value of this tension is selected from the range 25 ... 33% of the limit force for the spring, at which the stresses are close to the elastic limit. Such winding is called closed.

For cold-wound springs, annealed or pre-prepared material is used. In the case of using an annealed material, the springs are hardened and tempered after winding.

When using pre-prepared material (mainly carbon steels), only spring tempering is applied after winding. There are three types of pre-prepared material: cold rolled, hard drawn, heat treated.

In the case of hot winding, the pre-heat treatment of the workpiece is not very important. After the springs are manufactured by hot winding, the springs are heat treated.

The figure below shows a drawing of a compression spring with a spring characteristic (dependence of spring displacement on load).

Figure - Compression spring characteristic and technical requirements (example of a compression spring drawing) 

Figure - Drawings of tension and compression springs 

✍ On a note! The shape of the tension spring hooks is not standardized.

The figure below shows a drawing of a compression spring with a spring characteristic (dependence of spring displacement on load).

The figure shows a collage with drawings of springs and various options for hooks (trailers) of tension springs. Variants a and b are formed by bending the extreme turn, which is recommended for springs with a diameter of up to 8 mm with a corresponding decrease in the shear stress by 20 - 25%. Option d is an example of a tapered hook. Mounting on metal plates, options e and f, recommended for wire springs with a diameter of 0.2 to 5 mm.

Drawing - Spring hooks

Table. Shear modulus G for various materials

Note.

1 kgf/mm² = 9,81 MPa. For example, for spring steel   G = 8 · 10³kgf/mm² = 7,85 · 10⁴ MPa.

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