Numerical solution of Rayleigh equation in non‐linear vibration

Abstract

SUMMARYNumerical solution of the Rayleigh equation in non-linear vibration is studied in this paper. The di erential equation is integrated on a particular interval (0; T p2 ) with the initial value condition, u = A i and du=dt = 0 at the time t = 0. The value T p2 is determined from the condition such that the trajectory of motion on the phase plane is a unclosed path around the original point with the both starting and the end point on the positive real axis. The target function method is developed to obtain the particular value T p2 . The obtained A i+1 (= u(T p2 )) will be used in the initial value condition of the next round integration. A stable periodic motion is obtained after some rounds of integration. The solution technique is out of the small parameter assumption in the Rayleigh equation. Finally, numerical examples and results are given.

Dislocation-cracks interaction during fatigue: A discrete dislocation dynamics simulation

“…Recently, the screw dislocations around a radial surface crack in a circular bar 5 and extended dislocations around a semi-infi nite crack were investigated, 6 and the effect of fatigue on the failure of irradiated materials has been studied for copper alloys and pure iron. 7,8 Furthermore, discrete dislocation dynamics (DD) have been used to model the interaction between a sharp crack and dislocations.…”

“…Most of them use either boundary elements formulation 16,17 or semi-numerical methods. 18,19 The semi-analytical processes are very fast, but can only deal with cracks of specific shapes 19 and usually fail when the external traction is very complicated or changes very fast. The boundary element formulations are computationally …”

“…In this previous equation, is a characteristic crack length and d(x,y) is the distance of the center of each triangle from the crack face. The application of the above leads to Equation 6.…”

Mathematica solution of Rayleigh equation in non‐linear vibration

“…Chen and Lee [4] solve a series of problems for the amplitude. In every problem the corresponding period is computed as a root of target function.…”

“…Recently, in this journal Chen and Lee [4] ÿnd the stable periodic motion of Rayleigh equation, which appears after a su cient long time, by using the target function method [5][6][7][8]. This communication presents Mathematica [9] software system rules, computing the magnitude A, the periods T , the displacement u(t), and the velocity v(t) for the speciÿed perturbation parameter and circular frequency !.…”