Page 276 - Handbook of Materials Failure Analysis

272 CHAPTER 11 The reliability design of mechanical system

6.1 ACCELERATED FACTOR
First of all, to create a life-stress model and accelerated factor, time-to-failure can be
obtained from the McPherson’s derivation [10],

1 E a
TF ¼ B sinh aSðފ exp (11.2)
½
kT
where T f is time-to-failure, S is stress, E a is activation energy, k is Boltzmann
constant, T is absolute temperature, B and a are coefficients.
In medium stress, time-to-failure can be described as

E a
TF ¼ ASðÞ exp (11.3)
n
kT
The energy flow in a mechanical system can generally be expressed as efforts and
flows (Table 11.2). Thus, it is judged that the stresses in mechanical or electrical sys-
tem may come from the efforts (or loads) like [11].
For a mechanical system, when replacing stress with effort, the time-to-failure
can be modified as

E a E a
TF ¼ ASðÞ exp ¼ AeðÞ exp (11.4)
n
n
kT kT
The more severe the conditions, the shorter the testing time can be. This is a key fac-
tor of an accelerated life testing. In accelerated testing, the appropriate stress levels
(S 1 or e 1 ) will fall outside the specification limits but inside the operating limits.
Under accelerated stress conditions, the AF can be described as:

n n
S 1 E a 1 1 e 1 E a 1 1
AF ¼ ¼ (11.5)
S 0 k T 0 T 1 e 0 k T 0 T 1
We can say that the first term is the outside effort (or load) and the latter is the internal
energy in Equation 11.5. The equation has three unknown values—constant A and
two parameters which are temperature and efforts. These parameters can be found
using a three-level test under accelerated conditions. Under severe conditions, repet-
itive stress (or load) can be expressed as the duty effect that carries the on/off cycles
and shortens unit lifetime [12]. Now, we derive the sample size equation to perform
the parametric ALT.

Table 11.2 Effort and Flow in the Multi-Port System

Units Effort, e(t) Flow, f(t)
Mechanical Force component, F(t) Velocity component, V(t)
translation
Mechanical rotation Torque component, τ(t) Angular velocity component, V(t)
Compressor Pressure difference, ΔP(t) Volume flow rate, Q(t)
Electric Voltage, V(t) Current, i(t)