<p>pBifur Draghilev method</p>

d453ee08-0f5c-484c-9755-73d72091d335 241 4 5 false false 0 decimal

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Graph of implicit function with bifurcation point (Problem on the calculation of the arch)

The graph consists of three curves that occur at the bifurcation pont.

The curved arch , the ends of which are secured against movements along the x axis, is loaded uniform vertical load q. N-longitudinal force at the ends of the arch, E, F, J-module elasticity, area and moment of inertia of the arch cross section, h-boom lift, L-distance between supports.It is known that the relationship between load q and thrust force N has the form of a loop-shaped. The intersection point on the curve is a bifurcation point.Find dependence of N on q.

<p>pBifur Draghilev method</p>

AlgLib Solver for a system of nonlinear equation pBifur k 1 x f1 rows range u k el x k el : for x Jac u f1 u Jacobian : StepMax 0 : Eps 1014 -^: X0 X01 StepMax Eps x f1 x Jac al_nleqsolve : 51 line :

Draghilev method 2 f X0 tmin tmax N D J ξ F ξ 1 el diff ξ F ξ 2 el diff 12 mat : t ξ D1 J 2 el - J 1 el 12 mat transpose J norme / : X0 tmin tmax N t ξ D1 rkfixed 31 line :

The dependence of N on q satisfies the following equation:

ff x 2 el sqrt 2 / cos (2^π 2^x 2 el * 2 π 2^* x 2 el - (* 0.2 x 2 el * π 2^x 2 el - (2^* + (* : x F x 1 el (2^π 2^x 2 el - (2^* 6 x 2 el (3^* / 24 x 2 el + (x 2 el sqrt 2 / cos (2^* 615 x 2 el sqrt sin x 2 el sqrt / * - (* + (* 8 π * x 1 el * x 2 el sqrt 2 / cos (2^* - ff + : 21 sys

It is required to plot the function F (x)

Solution

1. Find the coordinates of the bifurcation point

x f1 x F x 1 el diff x F x 2 el diff 21 mat :

Vector of guesses X01 751012 mat transpose :

pBifur 76.5039 9.8697 21 mat

1

2.The starting point for each of the three graphs is taken near the bifurcation point

X1 771021 mat :

X2 77 9.8 21 mat :

X3 73.8 9.72 21 mat :

3. Build graphics by Draghilev method

tmax 43175 - 7031 mat :

N 4311514031 mat :

1 t x3 2 t x3 1 t x1 2 t x1 1 t x2 2 t x2 1 t x3 2 t x3 4 2 mat 1 t x1 2 t x1 1 t x4 2 t x4 2 2 mat cil1 B1γ transpose 1 el B1γ transpose 2 el 1 t x4 2 t x4 2 2 mat stack 1 t x1 2 t x1 . 15 brown 1 5 mat 1 t x2 2 t x2 . 15 brown 1 5 mat 1 t x3 2 t x3 . 15 brown 1 5 mat C1γ transpose 1 el C1γ transpose 2 el C2γ transpose 1 el C2γ transpose 2 el C3γ transpose 1 el C3γ transpose 2 el C1γ transpose 1 el C1γ transpose 2 el 4 2 mat 1 t x2 2 t x2 1 t x5 2 t x5 2 2 mat cil2 B2γ transpose 1 el B2γ transpose 2 el 1 t x5 2 t x5 2 2 mat stack t C1 t C2 . 15 green 1 5 mat 1 t x4 2 t x4 . 20 1 5 mat 1 t x5 2 t x5 . 20 1 5 mat 1 t x3 2 t x3 1 t x6 2 t x6 2 2 mat cil3 B3γ transpose 1 el B3γ transpose 2 el 1 t x6 2 t x6 1 t x3 2 t x3 3 2 mat stack 1 t x6 2 t x6 . 20 1 5 mat Sy γ * 1 col Sy γ * 2 col augment XYZ t C1 t C2 t C1 t C2 0.1 - 2 2 mat TextXYZ t 4 < path3 t C1 t C2 0.1 - . 13 green 1 5 mat : t 0 ≡ path3 t C1 t C2 0.1 - . 13 green 1 5 mat : path3 path3 t C1 t C2 0.1 - . 13 green 1 5 mat stack : if if 21 1 sys

верхняя кривая B1 x y f X1 0 tmax 1 el N 1 el D 1 N 1 el 23 submatrix :

правая кривая B2 x y f X2 0 tmax 2 el N 2 el D 1 N 2 el 23 submatrix :

B3 x y f X3 0 tmax 3 el N 3 el D 1 N 3 el 23 submatrix :

B1 B2 B3 3 1 sys B1 B2 B3 pBifur 1 el pBifur 2 el . 25 red 1 5 mat 4 1 sys