Stiffness of Spring Units

When designing furniture for lying down of sitting (with sleep and relaxation functions), anthropometric and physiological rules should be taken into account arising from their use. Mattresses, especially those of orthopaedic character, con­stitute one of the essential factors of the quality of life for people with musculo­skeletal dysfunction. Rehabilitation of such patients is an ongoing process, and the level of daily activity and wellness determines the behaviour of previously achieved effects or conditions progress in the improvement of abilities. A different per­spective of orthopaedics on the cause of health ailments caused by badly designed and manufactured upholstered furniture makes it that most mattresses are designed by intuitively selecting both the spring materials and the shapes and dimensions of beds and seats. Such approach to design, results of a number of inconsistencies between the requirements and expectations of users and normative recommenda­tions. Therefore, engineering design methods of spring methods are looked for in order to determine the most favourable material and construction parameters which improve ergonomics, and thus the user’s comfort of sleep and relaxation.

The following analysis concerns states established for objects, which is a system made up of cylindrical upholstery springs with: height H =125 mm, coil diameter D = 60 mm, wire diameter d =2.1 mm, the number of active coils n = 5 and linear elasticity modulus E =2 x 105 MPa (Fig. 8.38).

In order to determine the deformation of the entire spring system, under the influence of the load of concentrated force, first the behaviour of single springs subjected to compression and deflection is analysed (Fig. 8.39), since these deformations make up the deformation of the entire system of springs connected in parallel.

Finally, the stiffness of the bent spring was written as:


Gg = E/2(m + 1),

v Poisson’s ratio

The non-axial load of a single spring, however, causes the stiffness of it alone and the system of connected springs may be changed. Figure 8.40 presents the deflections of springs compressed eccentrically. It was assumed that the stiffness k of each spring can be expressed as the stiffness of the parallel system of springs of the stiffness k = 0.5 k1. As a result of such compression, the deflection of springs under the influence of individual loads has the form:

For the axial compression of the spring

, „ 1 P

F = F = 2 P, hence / = —, (8.158)

for the compression of the spring along the peripheral of the cylinder


F1 = 0, F1 = P, hence/1 = , (8.159)


for the compression of the spring with force applied at any point on the surface indicated by the passive coil

8.6 Stiffness of Spring Units and

/ _ 8РРщ x2 1 G1 df x1 + x2 ’

/1 = Uixi;


For the parallel connection of two springs with their peripherals, the deflection value of the set should depend on the dimensions of the springs and their stiffness. The deformation of the system presented in Fig. 8.42 results from the deflection

work of the spring set on the right side.

In this case, it can be written that

P = F1 + F" + F2 + F", (8.172)

1 3

2 F D = f’0d + – F2D, (8.173)

P = 2F1′ + 3F2′ + 2F1′ + F2′ = ^F1′ + F’D, (8.174)

1P = F1’+ F2′, (8.175)


3 P = F + F2, (8.176)

Fig. 8.42 Load scheme of springs connected by peripherals


Because comfortable use of a mattress of an upholstered furniture piece should be associated with an even distribution of stresses of the body on the spring layer of the furniture piece, when modelling the stiffness of the spring unit, both the effect of compression and deflection of individual springs should be taken into account. Therefore, the springs have been supported immovably on a stiff base, while on the ends, at the site of mutual contact, they were connected articulately (Fig. 8.43).

At constant parameters k1 and kg, characterising the stiffness of the springs, the stiffness of the considered unit will depend, among others, on the diameter D. The reaction of the unit can therefore be expressed as:


Mx, My appropriate bending moments (Fig. 8.44),

w(x, y) a function that describes the deflection of the unit surface caused by concentrated force

By specifying the deflection value of the spring unit, the principle of minimum potential energy was used. During compression with concentrated force P (Fig. 8.45) of the system in point A(xp, yp), the work of external forces on the external displacements is equal to the sum of working internal forces and potential energy of the deformed surface. The potential energy V of the whole system has been written in the form:

V = Uz – Uw, (8.186)

where the work of external forces:

Uz = 2 Pw(x, y), (8.187)

Fig. 8.44 Spring load by concentrated force and bending moments

Fig. 8.45 Loading the mattress surface by concentrated force in point A

and the work of internal forces:

hence finally

The parameters A i n occurring in the equation of deflection of the surface of the unit have been determined from the conditions:




dV д g

taking into account that the value of the deflection amplitude will not depend on the coordinates of the load point, the following was established:

In this way, the maximum deflection of the spring unit, made up of cylindrical springs connected articulately in the upper coils and loaded by concentrated force in the point, can be written in the form: