大门|大门-CS-2016（第1组）|问题10

在一个过程中，随着载荷的增加，失效循环次数呈指数下降。在80个单元的负载下，故障需要100个周期。当荷载减半时，需要10000个循环才能失效。在5000次循环中发生故障的载荷为。

（A） 40

（B） 46.02 （C） 60.01 （D） 92.02 答复: （B） 说明：

For exponential dependence we must have functions of form
f(x) = a.(k power x) : a,k are constants and f(0)=a

Let us assume: C = cycles for failure ; L = Load; a, k = constants
according to question C is an exponential function of L
Hence we can say,

1) C = a x (k^L) ---- > in case C is increasing exponentially
2) C = a / (k^L) ---- > in case C is decreasing exponentially

we take 2nd equation and apply log to both side we get:

log C + L x (log k) = log a ...... by logarithmic property

given (C=100 for L=80) and (C=10000 for L=40)
apply these to values to above equation, this gives us 2 equations
and 2 variables:

log a = log 100 + (80 x log k)...............(1)
log a = log 10000 + (40 x log k)...........(2)

we have 2 variables and 2 equations hence we find
log a = 6
log k= 1/20

now find, [ L = (log a - log C) / (log k) ] for C=5000 cycles

=> L = (6 - log 5000)*20 = 46.0206 ~ 46.02 => [ANS]

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