Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 162

Free end:
u(0,t)/ x = 0 (4.2.106)

Mass loaded end:
m

2 u(0,t)/ t 2 = AE u(0,t)/ x

(4.2.107)

Spring loaded end:
u x + u = 0

(4.2.108)

Dashpot end:
u x + u t = 0

(4.2.109)

4.2.10 Example Fortran implementation.
This chapter deals with drillstring vibrations; in particular, axial, lateral,
torsional and fully coupled vibrations, in that order. The numerical algorithms
discussed have been extensively tested, but we defer sample numerical
calculations until later in the discussion. Then we will describe a fairly general
Fortran program developed for coupled vibrations, expanding on that of Chin
(1988a,b). The research algorithm allows users to decouple or couple any
component modules as desired and to introduce general rock-bit boundary
condition models as needed.

4.2.10.1 Code fragment.
The code fragment shown in Figure 4.2.8 was extracted from this program
to illustrate how our difference equations are coded and solved by the
tridiagonal matrix solver in Figure 4.2.6. In this program, the following
nomenclature applies (for now, disregard , V and W).

... Product of area and Young's modulus

AE

... Damping factor in surface cables

BETA

... u(0,t)/ x at bit

DUDX

... Young's modulus ELAST

... Acceleration due to gravity

G

... Damping factor in drillstring

GAMA

... Power at bit

POWER

... " " rock-bit interaction parameter

RBALPH

... " " rock-bit interaction parameter

RBBETA

... " " rock-bit interaction parameter

RBLAMB

... Mass density

RHO

... Surface spring constant SPRING

... Mass of traveling block system

TBMASS

... u 0

UZERO

... Velocity at bit

VEL

...

WMEGA