Mathematical Physics - Volume II - Numerical Methods

3.1 Approximate analytical solution

51

and

2 "

1 2 (

4 − β 4 ) − β 4 # = 0 .

L

4 π

4 + C

− C 1 β

(3.15)

In order for the non-trivial solution of systems (3.14) and (3.15) to exist, their determinants must be equal to zero, from which it follows that: β 1 L = 4 , 74 β 2 L = 11 , 14 (3.16)

L 2 r L 2 r

22 , 48

EI m EI m

ω 1 =

(3.17)

124 , 1

ω 2 =

.

3.1.1 Finite difference method solution

Finite difference method requires approximate expressions for derivatives, which can be obtained via Taylor series:

d 2 f dx 2 d 2 f dx 2

( ∆ x ) 2 2 ( ∆ x ) 2 2

d 3 f dx 3 d 3 f dx 3

( ∆ x ) 3 6 ( ∆ x ) 3

d 4 f dx 4 d 4 f dx 4

( ∆ x ) 4 24 ( ∆ x ) 4 24

d f dx d f dx

f ( x + ∆ x ) = f ( x )+

∆ x +

(3.18)

+

+

f ( x − ∆ x ) = f ( x )+

∆ x −

(3.19)

6 −

+

.

Taking into account the first two terms and subtracting of equations (3.18) and (3.19) results in:

f ( x + ∆ x ) − f ( x − ∆ x ) 2 ∆ x .

d f dx x ≈

(3.20)

Taking into account the first three terms and subtracting of equations (3.18) and (3.19) results in:

d 2 f dx 2

f ( x + ∆ x ) − 2 f ( x )+ f ( x − ∆ x ) ( ∆ x ) 2

(3.21)

=

x

d 2 f dx 2

If f ( x ) in equation (3.21) is replaced with

x , the following is obtained:

d 2 f dx 2

d 2 f dx 2

d 2 f dx 2

2

x + ∆ x −

+

d 4 f dx 4

x − ∆ x

x

(3.22)

=

( ∆ x ) 2

x

By replacing (3.21) into the right-hand side of equation (3.22), we obtain: d 4 f dx 4 ≈ 1 ( ∆ x ) 2 f ( x + 2 ∆ x ) − 2 f ( x + ∆ x )+ f ( x ) ( ∆ x ) 2 − − 2 f ( x + ∆ x ) − 2 f ( x )+ f ( x − ∆ x ) ( ∆ x ) 2 +

(3.23)

+ f ( x ) − 2 f ( x − ∆ x )+ f ( x − 2 ∆ x ) ( ∆ x ) 2