Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 165

C(1) = +RBALPH/DX
W(1) = +RBBETA*UNM1(1)/DT
215 XDERIV = (UNM1(2)-UNM1(1))/DX
235 IF(VEL.GT.0.) GO TO 240
A(IBIT) = 1.
B(IBIT) = -2.
C(IBIT) = 1.
W(IBIT) = -UZERO*SIN(WMEGA*T)
A(IBIT+1) = 1.
B(IBIT+1) = -2.
C(IBIT+1) = 1.
W(IBIT+1) = +UZERO*SIN(WMEGA*T)
240 CALL TRIDI(A,B,C,VECTOR,W,IMAX)
DO 250 I=1,IMAX
UN(I) = VECTOR(I)
250 CONTINUE
POWER = -AE*VEL*XDERIV
DO 270 I=1,IMAX
UNM2(I) = UNM1(I)
UNM1(I) = UN(I)
270 CONTINUE
DUDX(1) = (UN(2)-UN(1))/DX
DUDX(IMAX) = (UN(IMAX)-UN(IMAXM1))/DX
DO 275 I=2,IMAXM1
DUDX(I) = (UN(I+1)-UN(I-1))/(2.*DX)
275 CONTINUE
C

Figure 4.2.8. Axial vibration code listing.
4.2.10.2 Modeling dynamically steady problems.
Drillstrings undergo fully transient vibrations, e.g., bit bounce, rate-of-
penetration, high shock loadings, intermittent drillbit chatter, and so on. But it
may well be that periodic external excitations do lead to periodic motions for
limited ranges in time. When a drillstring is excited harmonically at constant
frequency and periodicity is observed, the drillstring motion is “dynamically
steady” because its standing wave pattern remains unchanged with time. If such
a situation exists, it will be computed by the transient algorithm; if it does not, it
won't. Thus, a transient formulation is more general and less restrictive; for the
general algorithm in this chapter, personal computers more than suffice for the
most demanding computations. This, again, is so for two reasons. First, “elastic
line” modeling reduces the number of unknowns; second, the absolute stability
and second order spatial accuracy of our axial, torsional and lateral vibration
algorithms allows us to select large time steps.
Therefore, we will not formulate special dynamically steady models in this
topic. We will, however, review the basic analytical ideas, if only to
demonstrate the mathematics (and complications) involved. Let us consider
uniform drillstrings satisfying the governing partial differential equation
2 u d (x,t)/ t 2 +

u d (x,t)/ t - E 2 u d (x,t)/ x 2 = 0 (4.2.110)