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Steering of Vehicles

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Spring Suspension of Vehicles

With the construction of large and heavy vehicles it is to notice, that the weight should be spreaded evenly as possible onto all wheels to avoid overload of one wheel resp. of itīs suspension. At multi-axle vehicles it is also to ensure that the driving axle always has contact to the roadway.
Either can be realized by spring suspension; at this the necessary springs mostly are determined after completion by way of an experiment, what possibly requires an assortment of springs.
But there is another method; with some knowledge of springs their properties can be determined quite good and the suitable springs can be choosen selective.

The proper calculation of springs is difficult. The reason is, thatīs not possible to combine the spring parameters into one equation, but several equations are necessary whose solutions require an iterative calculation.

For the springing of metal kit vehicles restrictions are acceptable for a simplified method to choose usable springs:
Generally cylindrical helical pressure springs from steel with linear characteristic (spring stiffness = constant = spring rate c) are used whose loading can be assumed as a static. The middle diameter of the springs mostly is between 8 and 12 mm with a wire diameter from 0,8 mm to 1,2 mm, which allows spring tensions of least 20 N.
With these an further simplifications the system of equations can be reduced to some formulas. As stated below, a spring suspension with helical pressure springs shall be "calculated ".


The most important specification is the weight, with which the spring static is loaded.
This e. g. can be calculated from the mass of the model:

FG = m · g
 FG = weight in N (Newton)
m  = mass in kg
g   = gravitational acceleration, ca. 10 m/s²

It shall be assumed, that a model has an evenly spreaded mass of 8 kg.
The resulting weigth is 80 N and with 4 wheels every wheel has to carry a weigth of 20 N.
Is furthermore assumed, that the suspension of every wheel contains 2 pressure springs, every spring is loaded with 10 N.
Most models will not have an evenly spreaded mass. In this case the weight can be measured by lifting up the model above the affected axle with a spring balance; following the weight is to devide through the number of effective springs. At a twin axis it should be measured between the axles.


Another specification is the max. displacement f2.
Here it means the difference between the length of the unburdened spring L0 and the length of the complete compressed spring LBl. Also this is a simplification because a proper calculated spring of course not will be loaded up to the limit.
f2 is assumed to 20 mm.

If the model stands on a flat bottom, in this example the springs should be compressed approx. 10 mm, so they can move in both directions each with 10 mm.
The correlation between force effect and elongation is given by the so called spring rate c:

c = F / f
 c = spring rate in N/mm
F = to the spring effecting force in N
f = elongation in mm

In the actual case the required spring rate is c = 10 N / 10 mm = 1 N/mm. A complete compression of the spring accordingly requires a force of 20 N.


Last the length of the complete compressed spring LBl and the diameter can be provided.
But for the calculation not the outer diameter but the medial diameter Dm is necessary; so the outer diameter will be larger by the wire diameter, which is still to calculate. LBl is assumed to 10 mm and Dm also to 10 mm.


Under the premise that the spring consits only of effective windings (explanation follows) from the length of the complete compressed spring the product of the number of effective windings if and wire diameter d can be calculated:

if · d = LBl = 10 mm
if     = effective number of turns
d    = wire diameter in mm
LBl = length compressed in mm


Now only a "formula" is missing, which describes the correlation between all known and still unknown parameters. For this the equation
c = G · d4 / 8 · if · Dm³
c    = spring rate in N/mm
G   = torsion modulus, for steel 83000 MPa (N/mm²)
if    = number of effective turns
d    = wire diameter in mm
Dm = medial spring diameter in mm

can be used.
This equation still contains 2 unknown terms, which are involved by the input requirement d · n = 10 mm; so n can be replaced by 10 mm/d and following the equation can be solved to d:

d = 5th root (c · 80 mm · Dm³ / G) = 1 mm

For this example thus 8 steel springs are suitable with a medial diameter of Dm = 10 mm, a wire diameter of d = 1 mm and an effective number of turns of if = 10.

As crosscheck this values can be inserted into the above equation to control the spring rate.
For this in www.kreuzotter.de/deutsch/feder.htm exists a template, in which the separate parameters can be entered and the spring rate can be calculated "at the push of a button".

Of course the required spring rate also can be realized with other values; that depends on the input parameters. But at least this number of windings is necessary that the presumed displacement (here 20 mm) can be reached.


Important, but problematical is the length of the unburdened spring L0. It depends on the pitch of the spring and can be variably from one spring to another.
In practice L0 is the 3 to 5fold of LBl and can be 30 to 50 mm in this example. Therefore an adequate installation space is to be provided.


Finally to the number of effective turns:
A helical pressure spring usually has a cylindrical shape, that means that approx. the last 3/4 turn is to be pressed to the neighbouring turn and cannot be used for springing.
To reach the necessary number of effective turns approx. 1 to 2 turns more should be choosen; both the length of the complete compressed spring and the length of the unburdened spring will be longer as presumed.


Spring rates of current steel helical pressure springs in N/mm
if = number of effective turns     Dm = medial spring diameter in mm     d = wire diameter in mm

if
Dm=12
Dm=10
Dm=8
 
d=0,8
d=1,0
d=1,2
d=0,8
d=1,0
d=1,2
d=0,8
d=1,0
d=1,2
5
6
7
8
9
10
11
12
13
14
15
0,49
0,41
0,35
0,31
0,27
0,25
0,22
0,20
0,19
0,18
0,16
1,20
1,00
0,86
0,75
0,67
0,60
0,55
0,50
0,46
0,43
0,40
2,49
2,08
1,78
1,56
1,38
1,25
1,13
1,04
0,96
0,89
0,83
0,85
0,71
0,61
0,53
0,47
0,43
0,39
0,35
0,33
0,30
0,28
2,08
1,73
1,48
1,30
1,15
1,04
0,94
0,86
0,80
0,74
0,69
4,30
3,59
3,07
2,69
2,39
2,15
1,96
1,79
1,65
1,54
1,43
1,66
1,38
1,19
1,04
0,92
0,83
0,75
0,69
0,64
0,59
0,55
4,05
3,38
2,89
2,53
2,25
2,03
1,84
1,69
1,56
1,45
1,35
8,40
7,00
6,00
5,25
4,67
4,20
3,82
3,50
3,23
3,00
2,80

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