A Sixth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems

We present a sixth-order finite difference method for the generalsecond-order non-linear differential equation Y"=f(x, y, y')subject to the boundary conditions y(a) = A, y(b) = B. In thecase of linear differential equations, our finite differencescheme leads to tridiagonal linear systems. We establish, underappropriate conditions, O(h⁶)-convergence of the finite differencescheme. Numerical examples are given to illustrate the methodand its sixth-order convergence.     

DYNAMICS OF A TWO-CRACK ROTOR

The effect of the presence of the single transverse crack on the response of the rotor has been a focus of attention for many researchers. In the present work a simple Jeffcott rotor with two transverse surface cracks has been studied. The stiffness of such a rotor is derived based on the concepts of fracture mechanics. Subsequently, the effect of the interaction of the two cracks on the breathing behavior and on the unbalance response of the rotor is studied. When the angular orientation of one crack relative to the other is varied, significant changes in the dynamic response of the rotor are noticed. A special case of practical importance of a two-crack rotor is one when one of the cracks is assumed to remain open always whereas the other can breathe like a fatigue crack. This simulates a transverse crack in an asymmetric rotor. Effect of orientation of the breathing crack with respect to the open crack on the dynamic response is studied in detail. The results of the present study will be useful in diagnosing fatigue cracks in real rotors, which invariably have some asymmetry.     

A New Fourth-order Finite-difference Method for Computing Eigenvalues of Fourth-order Two-point Boundary-value Problems

We present a new finite-difference method for computing eigenvaluesof two-point boundary-value problems involving a fourth-orderdifferential equation. Our finite-difference method leads toa generalized seven-diagonal symmetric-matrix eigenvalue problemand provides O(h⁴)-convergent approximations for the eigenvalues.     

ANALYSIS OF THE RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL EXCITATION

The coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. Steady state unbalance response of a Jeffcott rotor with a single centrally situated crack subjected to periodic axial impulses is studied. Partial opening of crack is considered and the stress intensity factor at the crack tip is used to decide the extent of crack opening. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Therefore, lateral vibration response of a cracked rotor to axial impulses is studied in detail. Spectral analysis of response to periodic multiple axial impulses shows the presence of rotor bending natural frequency as well as side bands around impulse excitation frequency and its harmonics due to modulations caused by rotor running frequency. It is concluded that the above approach can prove to be a useful tool in detecting cracks in rotors.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

High-accuracy P-stable Methods for y' = f(t, y)

We obtain a one-parameter family of sixth-order P-stable methodsfor the numerical integration of periodic or near-periodic differentialequations that are defined by initial-value problems of theform: y^" = f(t, y), y(t₀)= y₀, y^'(t₀)= y^’₀. Our P-stablemethods are symmetric and involve three function evaluationsper step (periteration, in case f(t, y) is non-linear in y).For non-linear problems, starting values for the solution ofthe implicit equations by modified Newton's method are suggestedand illustrated by an example.     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.