BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 97
that in the network nodes would be fulfilled the relation: f(l i ) = ξ i+1 = ξ i+1 (l i );
L
i= 0, n. It must be mentioned that l i
= i ; i= 0, nand l 0 = 0; l n = L.
n
The classic algorithms for function interpolation upon exact data defined
on a discreet lot of points are based on interpolative polynoms (for example,
Lagrange, Newton, Stirling, Bessel polynoms).
The use of these methods is difficult, due to some particularities in the
field of application, and also due to the interpolation polynoms grade, equal to
the n number of intervals. In such situation, the procedure increase in accuracy
would be with a very complex mathematical device. For the simplicity was
preferred an interpolation method with spline – function, which are – usually –
polynomial function on portions. Most used spline functions are III-rd grade
polynomial.
On the [0, L] segment of the real axis attached to vehicle, a network of n
equal intervals was build, separated by (n+1) equidistant points, in which nodes
are specified the values of the function ξ i (l i ), i= 0, n. In the network nodes the
L
“length” variable takes the values l i
= i , i= 0, n, with l 0 = 0; l n = L.
n
Cubic interpolation problem on portions requires the determination of a
function f:[0, L]→ϒ; f = f(l) which satisfies the conditions:
1) f(l) ∈ C 2 (0, L) that means that it is continuous along with it is II -nd
grade derivatives including, on the definition interval. This condition suits the
explicit interpretation of vehicles deformation.
2) on each of the equal segments of the definition interval, [l i-1 , l i ] ,
i= 1, n, the f(l) function is grade III polynom, with the form
3
i k
f () 1 = fi() 1 = ∑ ak ( li
−l)
, i = 1 , n ; (1)
k = 0
3) in the network nodes the equalities are fulfilled
( ) 1 ,
f li
ξ
i +
4) f // (l) satisfies the limit condition
= i = 1,
n ; (2)
( 0) ( )
= =0, i = 1,
n . (3)
// //
f f L
As the function f(l) second derivative is continuous and linear on each [l i--1 ,
l i ] , i= 1, n interval of the network, it can be written that, for l i-1 ≤ l ≤ l i ( i= 1, n)