Consider a cold$-$drawn AISI $1040$ flat leaf spring subjected to a fluctuating loading condition of maximum and minimum stresses of $420$ MPa and $140$ MPa respectively for a total of $50,000$ cycles.$$ The loading condition of the spring then changes to maximum and minimum stresses of $350$ MPa and $-200$ MPa respectively.$$ Estimate the remaining life of the spring under the new loading condition using Palmgren$-$Miner cumulative damage theory.$$ Assume that the fully$-$corrected endurance limit is $200$ MPa and Sn$$is $90\%$ of Sut.$$Then$,$ calculate the remaining life of the leaf spring using Manson's method for linear damage.