Design of metal springs – Part 2 “Calculation” › Gutekunst Federn › Berechnungsformeln, Federberechnung, Formeln, Schenkelfedern, Springs, Tension springs

In the first part of this two-part series has Gutekunst feathers about the Basics of spring design informed. In this second part you will find the specific calculation data for the design of Compression springs , Tension springs and Leg springs (Torsion springs). This is also available to you for individual calculation Gutekunst spring calculation program WinFSB to disposal.

The aim of the spring design of a compression spring, tension spring or leg spring is to find the most economical spring for the given task, taking into account all the circumstances, which also fits into the available space and which is required lifespan reached. In addition to these manufacturing and material requirements, there is also the right one Spring design particular importance to.

The designer should put together the following requirements:

1. Load type (static or dynamic)

2. lifespan

3. Operating temperature

4. Ambient medium

5. Necessary forces and spring travel

6th Existing installation space

7th Tolerances

8th. Installation situation (Buckling, lateral suspension)

Every spring design consists of two stages:

Proof of function : Checking the spring rate, the forces and the spring travel, the vibration behavior, etc.
Proof of strength : Check for compliance with the permissible stresses or proof of fatigue strength.

This requires an iterative approach.

The Proof of strength is based on the decision whether the spring is loaded statically, quasi-statically or dynamically. The following criteria should be used for delimitation:

Static or quasi-static stress : Time-constant (resting) load or time-variable load with less than 10,000 strokes in total.
Dynamic stress : time-varying loads with more than 10,000 strokes. The spring is usually pre-tensioned and subjected to periodic swell loads with a sinusoidal curve that occur randomly (stochastically), for example in vehicle suspensions. In some cases sudden changes in force occur.

When dimensioning the springs, stress limits are to be specified which are based on the Strength values of the materials and take into account the type of stress. A safety factor is included to determine the permissible voltage. After a comparison with the actual tension, the spring dimensioning must be revised using an iterative procedure. The following applies:

Nominal voltage ≤ permissible voltage

Calculation of compression springs

General

Cold formed cylindrical compression springs with constant incline are most commonly used in practice. The wire is cold formed by being wound around a mandrel. Depending on the advance of the pitch pin, the coil spacing and the position of the spring are regulated. After winding, tempering takes place in order to reduce internal stresses in the spring and to increase the shear elasticity limit. So the Setting amount . The tempering temperatures and times depend on the material; cooling takes place in air at normal room temperature.

Other important work steps in spring production are grinding and setting. The spring ends are usually ground from a wire thickness of 0.5 mm in order to guarantee a plane-parallel mounting of the spring as well as an optimal introduction of force.

Exceeds when the spring is loaded Shear stress the permissible value, a permanent deformation occurs, which manifests itself in the reduction of the unstressed length. This process is called “setting” in spring technology, which is associated with the terms “creeping” and “ relaxation “From materials engineering is to be equated. To counteract this, the compression springs are wound longer by the expected amount of setting and later compressed to block length. This setting enables a better material utilization and allows a higher load in later use.

Calculation formulas cylindrical compression spring

The calculation of the Compression spring based on the calculation equations from DIN EN 13906-1:

Image: Theoretical compression spring diagram

Proof of function of compression springs

The following applies to cylindrical compression springs made of wire with a circular cross-section:

Spring rate: $R=\frac{ Gd^<wpml_curved wpml_value='4'></wpml_curved>}{8D^<wpml_curved wpml_value='3'></wpml_curved>n}$

from R = F / s follows:

Spring force: $F=\frac{ Gd^<wpml_curved wpml_value='4'></wpml_curved>s}{8D^<wpml_curved wpml_value='3'></wpml_curved>n}$

such as:

Suspension travel: $s=\frac{8D^<wpml_curved wpml_value='3'></wpml_curved>nF}{Gd^<wpml_curved wpml_value='4'></wpml_curved>}$

Proof of strength compression spring

After the spring dimensions have been determined, the strength must be verified. To do this, the existing shear stress is determined:

Tension from power: $\tau=\frac<wpml_curved wpml_value='8DF'></wpml_curved>{\pi d^<wpml_curved wpml_value='3'></wpml_curved>}$

Tension out of way: $\tau=\frac<wpml_curved wpml_value='Gds'></wpml_curved>{\pi n D^<wpml_curved wpml_value='2'></wpml_curved>}$

While the shear stress τ is to be used for the design of statically or quasi-statically loaded springs, the following applies corrected shear stress τ_k for dynamically stressed springs. The distribution of shear stress in the wire cross-section of a spring is uneven, the highest stress occurs on the inside diameter of the spring. With the tension correction factor k, which depends on the winding ratio (ratio of mean diameter to wire thickness) of the spring, the highest tension can be approximately determined. For dynamically stressed compression springs the result is:

Corrected shear stress: $\tau_<wpml_curved wpml_value='k'></wpml_curved>=k\tau$

where for k applies (according to Bergsträsser):

$k=\frac{\frac<wpml_curved wpml_value='D'></wpml_curved><wpml_curved wpml_value='d'></wpml_curved>+0,5}{\frac<wpml_curved wpml_value='D'></wpml_curved><wpml_curved wpml_value='d'></wpml_curved>-0,75}$

Now the comparison is made with the permissible voltage. This is defined as follows:

Allowable voltage:

$\tau_{<wpml_curved wpml_value='zul'></wpml_curved>}=0,5\cdot R_{<wpml_curved wpml_value='m'></wpml_curved>}$

and

$\tau_{<wpml_curved wpml_value='czul'></wpml_curved>}=0,56\cdot R_{<wpml_curved wpml_value='m'></wpml_curved>}$

The values for the Minimum tensile strength R_m are dependent on the wire thickness and can be found in the standards of the corresponding materials.

As a rule, it must be possible to compress compression springs up to the block length, which is why the permissible stress at block length is t_czul to consider.

In the case of dynamic loads Low and high tension (t_k 1 and t_k 2) of the corresponding stroke can be determined. The difference is the stroke voltage. Both the upper tension and the stroke tension must not exceed the corresponding permissible values. These can be found in the fatigue strength diagrams in EN 13906-1: 2002. If the stresses withstand this comparison, the spring is fatigue-resistant with a limit load cycle of 10^7th .

Geometric relationships in compression springs

Spring size	Calculation equation
Total number of turns	n_t = n + 2
Block length of the ground spring	L._c = n_t d_Max
Block length of the unpolished nib	L._c = (n_t + 1.5) d_Max
Smallest usable length	L._n = L_c + S_a
Unstrained length	L.₀ = L_n + s_n
Sum of the minimum distances between the turns	$S_<wpml_curved wpml_value='a'></wpml_curved>=\left (0,0015 \frac{D^<wpml_curved wpml_value='2'></wpml_curved>}<wpml_curved wpml_value='d'></wpml_curved> + 0,1d \right )\cdot n$
Enlargement of the outside diameter under load pitch	$\triangle D_<wpml_curved wpml_value='e'></wpml_curved>=0,1\frac{S^<wpml_curved wpml_value='2'></wpml_curved>-08Sd-0,2d^<wpml_curved wpml_value='2'></wpml_curved>}<wpml_curved wpml_value='D'></wpml_curved>$ $S=\frac<wpml_curved wpml_value='L0-d'></wpml_curved><wpml_curved wpml_value='n'></wpml_curved>$ (ground) $S=\frac{L0-2,5d}<wpml_curved wpml_value='n'></wpml_curved>$ (unpolished)
Buckling spring travel (valid for various Support coefficients n, see EN 13906-1: 2002)

All dynamically stressed springs with one wire size> 1mm should shot peened will. This increases the fatigue strength. After both the functional verification and the strength verification have been carried out, various geometry calculations must be carried out and taken into account in order to achieve the Feather fitting to be able to insert into the construction of the component. The block length can not be undercut, because the turns are tight against each other, the smallest usable length should not be undercut because then a linear force curve as well as dynamic resilience are no longer guaranteed. In addition, the permissible tolerances according to DIN 2095 must be taken into account.

Calculation of tension springs

General

Tension springs are wound around a mandrel just like compression springs, but with no distance between the windings and with different Eyelet shapes / Spring ends to attach the spring. The turns are pressed tightly against one another in terms of manufacturing technology. This inner Preload F₀ depends on the winding ratio and cannot be manufactured to any desired height. The provides reference values for the amount of preload Calculation software WinFSB of Gutekunst feathers after entering the respective spring data.

Image: Common eyelet shapes: a.) half German eyelet; b.) whole German loop; c.) hook eye; d.) English eyelet; e.) curled hook; f.) screw-in piece

The advantage of tension springs is that Freedom from kinks Disadvantages are the larger installation space and the complete interruption of the flow of force when the spring breaks.

Calculation formulas cylindrical tension spring

According to the calculation equations for compression springs, but taking the preload force into account, the following relationships apply to cylindrical tension springs made of round wire (see also Figure 1.8):

Image: Theoretical tension spring diagram

Proof of function of the tension spring

The following applies to cylindrical tension springs made of wire with a circular cross-section:

Spring rate: $R=\frac{Gd^4}{8D^3n}=\frac<wpml_curved wpml_value='F-F0'></wpml_curved><wpml_curved wpml_value='s'></wpml_curved>$

from R = F / s follows:

Spring force: $F=\frac{Gd^4s}{8D^3n}+F0$

such as:

Suspension travel: $s=\frac{8D^3n(F-F0)}{Gd^4}$

Proof of strength of tension springs

As with compression spring calculations, the existing shear stress must be determined.

Shear stress: $\tau=\frac<wpml_curved wpml_value='8DF'></wpml_curved>{\pi d^3}$

The corrected stroke tension must also be calculated for dynamic loads.

Corrected shear stress: $\tau_{<wpml_curved wpml_value='k'></wpml_curved>}=k\tau$

Allowable voltage: $\tau_{<wpml_curved wpml_value='zul'></wpml_curved>}=0,45 \cdot R_{<wpml_curved wpml_value='m'></wpml_curved>}$

The existing maximum voltage t_n for the greatest travel s_n is set equal to the permissible voltage. To however Relaxation To avoid this, only 80% of this spring travel should be used in practice.

$s_{<wpml_curved wpml_value='2'></wpml_curved>}=0,8 \cdot s_{<wpml_curved wpml_value='n'></wpml_curved>}$

For dynamic loads, no generally applicable Fatigue strength values must be specified, as the Bending points of the eyelets additional stresses occur, some of which can exceed the permissible stresses. Tension springs should therefore only be subjected to static loads if possible. If dynamic stress cannot be avoided, one should Eliminate curved eyelets and rolled or screwed-in end pieces insert e. A life test under later operating conditions makes sense. A surface consolidation through Shot peening is not feasible because of the tight turns.

Geometric relationships in tension springs

Spring size	Calculation equation
body length	L._K = (n_t + 1) d
Unstrained length	L.₀ = L_K + 2 L_H
Eyelet height half German eyelet	L._H = 0.55D_i up to 0.80D_i
Eyelet height whole German eyelet	L._H = 0.80D_i to 1.10D_i
Eye height hook eye	L._H> 1.10D_i
Eyelet height English eyelet	L._H = 1.10D_i

The permissible manufacturing tolerances according to DIN 2097 must be taken into account.

Calculation of torsion springs (torsion springs)

General

Spiral cylindrical Leg springs (Torsion springs) have essentially the same shape as cylindrical ones pressure – and Tension springs , but with the exception of the spring ends. These are bent in a leg shape in order to allow the spring body to rotate around the spring axis. This means that there are very many different areas of application, for example as return or hinge springs. The torsion spring should be mounted on a guide mandrel and the load should only be applied in the winding direction. The inside diameter is reduced here. The springs are usually coiled with no pitch. However, if friction is absolutely undesirable, torsion springs can also be manufactured with a coil spacing. In the case of dynamic loading, it must be ensured that there are no sharp-edged bends at the spring ends in order to avoid unpredictable stress peaks.

Calculation formulas for cylindrical torsion springs (Torsion springs)

The calculation is based on the guidelines of EN 13906-3: 2001:

Image: Theoretical torsion spring / torsion spring diagram

Proof of function of torsion springs (torsion springs)

Spring torque rate: $R_<wpml_curved wpml_value='M'></wpml_curved>=\frac<wpml_curved wpml_value='M'></wpml_curved>{\alpha}=\frac{d^4E}<wpml_curved wpml_value='3667Dn'></wpml_curved>$

Spring torque: $M=FR_<wpml_curved wpml_value='H'></wpml_curved>=\frac{d^4E\alpha}<wpml_curved wpml_value='3667Dn'></wpml_curved>$

Rotation angle: $\alpha=\frac<wpml_curved wpml_value='3667DMn'></wpml_curved>{Ed^4}$

Proof of strength of torsion springs (torsion springs)

The existing bending stress is determined and compared with the permissible stress. In the case of dynamic loading, the corrected stress must again be used for comparison.

Bending stress: $\sigma=\frac<wpml_curved wpml_value='32M'></wpml_curved>{\pi d^3}$

Corrected bending stress: $\sigma_{<wpml_curved wpml_value='q'></wpml_curved>}=q \sigma$

where for q applies:

$q=\frac{\frac<wpml_curved wpml_value='D'></wpml_curved><wpml_curved wpml_value='d'></wpml_curved>+0,07}{\frac<wpml_curved wpml_value='D'></wpml_curved><wpml_curved wpml_value='d'></wpml_curved>-0,75}$

Permissible bending stress: $\sigma_{<wpml_curved wpml_value='zul'></wpml_curved>}=0,7Rm$

In the case of dynamic loading, the lower and upper stress (t_k 1 and t_k 2) of the corresponding stroke can be determined. The difference is the stroke voltage. Both the upper tension and the stroke tension must not exceed the corresponding permissible values. For spring steel wire, these can be found in the fatigue strength diagrams in EN 13906-3: 2001. If the stresses withstand this comparison, the spring is fatigue-resistant with a limit load cycle of 10^7th .

Geometric relationships in torsion springs (torsion springs)

Spring size	Calculation equation
Reduction of the inside diameter at maximum load	$Di_<wpml_curved wpml_value='n'></wpml_curved>=\frac<wpml_curved wpml_value='Dn'></wpml_curved>{n+\frac{\alpha}<wpml_curved wpml_value='360'></wpml_curved>}-d$
Unloaded body length	$Lk=(n+1,5)d$
Body length in the maximally loaded condition	$Lk_<wpml_curved wpml_value='n'></wpml_curved>=(n+1,5+\frac{\alpha}<wpml_curved wpml_value='360'></wpml_curved>)d$
Suspension travel	$s_<wpml_curved wpml_value='n'></wpml_curved>= \frac{\alpha_<wpml_curved wpml_value='n'></wpml_curved>R_<wpml_curved wpml_value='H'></wpml_curved>}{57,3}$

In addition, the manufacturing tolerances according to DIN 2194 must be taken into account.

A summary of the article “Design of a metal spring”, consisting of Part 1 “Basics” and Part 2 “Calculation” can also be downloaded from the Gutekunst springs 1×1 .

Should you need one individual spring design just email us the key data for the metal spring you need technik@gutekunst-co.com , contact our technology department by phone at (+49) 035 877 227-11 or use at https://www.federnshop.com the Gutekunst spring calculation program WinFSB for free calculation of compression springs, tension springs and torsion springs.

For more information:

Compression spring design (video)

Extension springs design (video)

Design of metal springs – Part 1 “Basics”

Dimension data for spring design

Spring calculation program WinFSB 8

Spring steel wires and their properties

Design of metal springs – Part 2 “Calculation”