Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 232

EI (w i-2,n -4w i-1,n +6 w i,n -4w i+1,n +w i+2,n )/( x) 4 (4.5.5)

+ N i,n (w i-1,n -2w i,n +w i+1,n )/( x) 2

+ { ( N i+1,n - N i-1,n )/2 x}{(w i+1,n -w i-1,n )/2 x}

+ kw i,n + (w i,n -w i,n-2 )/2 t

+ A(w i,n -2w i,n-1 +w i,n-2 )/( t) 2

q (w) i,n-1
+T i,n (-v i-2,n-1 +2 v i-1,n-1 -2v i+1,n-1 +v i+2,n-1 )/2(

=

x) 3

+ { ( T i+1,n -T i-1,n )/2 x}(v i-1,n-1 -2v i,n-1 +v i+1,n-1 )/( x) 2

and

N i,n = - AE { u i+1,n - u i-1,n }/2 x (4.5.6)

Note that q (v) and q (w) need not be identical, and likewise, similar considerations
apply to the spring constant k and damping factor (since v and w are displaced
by 90 o , they “see” different sides of the hole). Also note that the Greek letters
used for lateral vibrations should not be confused with those used for axial and
torsional vibrations. Now, Equations 4.5.4 and 4.5.5 can be rearranged to
highlight their pentadiagonal structure as follows,

{EI/( x) 4 } v i-2,n (4.5.7)

+ {-4EI/( x) 4 + N i,n /( x) 2 - (N i+1,n -N i-1,n )/(4( x) 2 )} v i-1,n

x) 4 -2N i,n /(

+ { 6 EI/ ( x) 2 + k + /(2 t) + A/( t) 2 } v i,n

x) 4 + N i,n /( + {-4EI/( x) 2 + ( N i+1,n -N i-1,n )/(4( x) 2 )} v i+1,n

+ { EI/ ( x) 4 } v i+2,n
= (v) i,n-1
- T i,n (-w i-2,n-1 +2 w i-1,n-1 -2w i+1,n-1 +w i+2,n-1 )/{2( x) 3 }

- {(T i+1,n - T i-1,n )/(2 x)}{(w i-1,n-1 - 2w i,n-1 + w i+1,n-1 )/( x) 2 }

+ ( /(2 t))v i,n-2 + { 2 A/( t) 2 }v i,n-1 - { A/( t) 2 }v i,n-2

x) 4 } w i-2,n {EI/( (4.5.8)

x) 4 + N i,n /(

+ {-4EI/( x) 2 - (N i+1,n -N i-1,n )/(4( x) 2 )} w i-1,n

+ { 6 EI/ ( x) 4 -2N i,n /( x) 2 + k + /(2 t) + A/( t) 2 } w i,n

+ {-4EI/( x) 4 + N i,n /( x) 2 + ( N i+1,n -N i-1,n )/(4( x) 2 )} w i+1,n

+ { EI/ ( x) 4 } w i+2,n
= (w) i,n-1
+ T i,n (-v i-2,n-1 +2 v i-1,n-1 -2v i+1,n-1 +v i+2,n-1 )/{2( x) 3 }

x) 2 } + { ( T i+1,n - T i-1,n )/(2 x)}{(v i-1,n-1 - 2v i,n-1 + v i+1,n-1 )/(

+ ( /(2 t))w i,n-2 +{ 2 A/( t) 2 }w i,n-1 -{ A/( t) 2 }w i,n-2