A relay arm (Figure 34.1) is both an elastic beam and a mini-busbar — that is, a current-carrying conductor in which the i2R losses must be kept small. The response time of the relay is limited by the lowest natural vibration frequency of the beam, and the dimensions of the beam are also constrained by the requirement that a given opening force, F, can deflect it by the (given) opening deflection, d — that is, its stiffness is prescribed. Furthermore, the beam must not fail by fatigue. We wish to achieve all this while minimizing its electrical resistance. The design requirements are listed in Table 34.1
Figure 34.1 A relay arm. It must respond quickly, withstand fatigue loading, and conduct well.
FUNCTION |
Relay arm |
|---|---|
OBJECTIVE |
Minimize resistive losses |
Minimize response time |
|
CONSTRAINTS |
(a) Must not fail in fatigue after > 108 cycles |
(b) Length L and opening displacement d specified |
Table 34.1 The design requirements
The arm is modeled as a cantilever beam of length L and section bh (Figure 34.1). The maximum value of width b is fixed by space constraints. The resistive loss when a current i passes along the arm is
 , |
(M34.1) |
where ρe is the electrical resistivity of the material of which the arm is made. This is the quantity to be minimized: it is the objective function.
There is a constraint: that of fatigue life. The relay arm, when in the open position, suffers an end-displacement δ, caused by the opening force F. This generates a maximum surface stress in the beam of
 , |
(M34.2) |
where I = bh3/12 is the second moment of area of the beam. Force F is related to δ by
 , |
(M34.3) |
where C1 is a constant. (For an end-loaded cantelever, C1 = 3.) Eliminating I by substituting equation (M34.3) into (M34.2), and requiring that the stress σ is always less than the endurance limit, σe, gives:
 . |
This gives an equation for the aspect ratio, h / L of the arm:
 . |
(M34.4) |
Substituting equation (M34.4) into the objective function (M34.1) gives
 . |
(M34.5) |
For a given length L and opening, δ, the resistive loss is minimized by maximizing
 . |
The second objective is that of minimizing response time. The lowest natural flexural vibration frequency of the cantilever is
 |
(M34.6) |
where ρ is the density of the material of the beam and C2 is a constant. (For a cantilever, C2 = 0.56.) The response time is limited by the lowest natural frequency, so we wish to maximize f. Inserting I = bh3/12 and replacing h/L by the constrained value of equation (M34.4) gives
 . |
The frequency is maximized by maximizing
 . |
This is the index for light springs.
Figure 34.2 is a chart with M1 and M2 as axes, constructed using the branches of the materials tree involving conductors. The selection depends on the relative importance of the resistive loss and the response time, but it is reasonable, with no further information on this issue, to examine the materials which maximize both: they lie at the top right, and include those listed in Table 34.2. The copper-based composites Cu-Ag(f), Cu-Nb(f) and Cu-Al2O3(p) are excellent. Copper-beryllium alloys and phosphor bronzes have outstanding mechanical properties (M2) but conduct less well.
Figure 34.2 A chart with axes of M1 and M2, constructed from the Conductors record subset.
MATERIAL |
COMMENT |
Copper-silver composites |
Outstanding M1 and M2; expensive |
Copper-niobium composites |
Good M1, outstanding M2; expensive |
Copper-Al2O3(p) composites |
Good M1, outstanding M2; excellent high-temperature performance; expensive |
Cu-Beryllium alloys |
Outstanding M1 and M2; less expensive than the composites |
Phosphor bronzes |
Lower performance than Cu-Be, but cheaper |
Table 34.2 Materials for relay arms
The relay arm is typical of applications in which high electrical conductivity is sought, with the ability to carry repeated flexural loads. Flexible current-loads, spring-contacts and snap-connectors all have this feature.
Boiten, RG (1963) 'The Mechanics of Instrumentation', Proc. I. Mech. E., 177, No 10, 269–288.