No abstract

No abstract

It is sparse and inconclusive that research on the subject whether the fatigue life of the structure will be reduced by shot peening strengthening before shot peen forming (S + F), and this study investigates accordingly. First, the crack growth rate test of the machine-processing plate and shot peening strengthening before shot peen forming plate demonstrate that both plates’ final crack growth rate and length are similar. However, the test shows the “fluctuation phenomenon” of crack growth rate and the “intersection phenomenon” in the Paris curve. This study is based on a self-developed simulation plugin for crack growth paths. The results verify that “fluctuation” causes the differential distribution of the overall stress intensity factor in the strengthened (4.5% increase compared to machine-processing) and formed (9.8% decrease compared to machine-processing) crater areas of the shot peening strengthening before shot peen forming plate. Comparing to the full coverage strengthening area, the forming area (only 30% coverage) in the early stage of growth as well as the gain amplitude of the residual stress in the late stage of growth gradually decrease and tend to be the same as that of the machine-processing, as validated by the “intersection phenomenon”.

Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from 1, 2, 3, 5, . . .. This has been generalized by many authors, in particular to constant coefficient fixed depth linear recurrences with positive (or in some cases non-negative) coefficients. In this work we extend this result to a recurrence with non-constant coefficients, an+1 = nan + an−1. The decomposition law becomes every m has a unique decomposition as siai with si ≤ i, where if si = i then si−1 = 0. Similar to Zeckendorf's original proof, we use the greedy algorithm. We show that almost all the gaps between summands, as n approaches infinity, are of length zero, and give a heuristic that the distribution of the number of summands tends to a Gaussian.Furthermore, we build a game based upon this recurrence relation, generalizing a game on the Fibonacci numbers. Given a fixed integer n and an initial decomposition of n = na1, the players alternate by using moves related to the recurrence relation, and whoever moves last wins. We show that the game is finite and ends at the unique decomposition of n, and that either player can win in a two-player game. We find the strategy to attain the shortest game possible, and the length of this shortest game. Then we show that in this generalized game when there are more than three players, no player has the winning strategy. Lastly, we demonstrate how one player in the two-player game can force the game to progress to their advantage.

No abstract