ON BIFURCATION OF BAR AXIS BY COMPLEX

RESISTANCE TO COMPRESSION AND BENDING

Let an initially straight control bar, one end of which is fixed and another free, be clamped by constant compressing force P at the free end and the constant distributed load of intensity g . In this case one has (see [1])

(1)   av'' - Q(s)sin v + (b²/Q²(s))sin 2v = 0, v(0)=v'(l)=0.

Here Q(s)º -(P₀+l ₁)+(g ₀+l ₂)(s-l); b=>0; v(s) is bar sections rotation angle; s is Lagrange coordinate, 0 £ s £ l; l - length of the bar; a is bending rigidity; b₁ is stretching rigidity; b₂ is shifting rigidity.

Put P=mg (m=const>0) and investigate the problem (1) near the values of parametra P=P₀ and g =g ₀. Clearly l ₁=ml ₂.

For (P₀,g ₀) to be a bifurcation point of the problem (1), it is necessary that the parameter g ₀ be an eigenvalue of the problem

(2)   av''+g ₀(m+l-s)v+g b(m+l-s)² v=0, v(0)=v'(l)=0.

The problem (2) has a sequence of real eigenvalues {g _0k}. They are all simple and |g _0n| ® +¥ , when n ® ¥ , and the corresponding eigenfunctions are

j k(s)=

K(s,t)ek(t)dt, k=1,2,...,

 

where K(s,t) is the Green function of the operator v'' under the boundary conditions v(0)=v'(l)=0, whereas e_k(s) (k=1,2,...) have certain extremal properties.

Using methods of branching theory [2] it is proved that the problem (1) has aside of the trivial solution two small real ones, defined only for l ₂>0. These solutions, expressing supercritical deformation of the bar, are given in form of convergent series by fractional powers of the parameter l ₂.

[1] V.V. Eliseev, On Nonlinear Dynamics of Elastic Bars; Prikl. Mat. Mekh., # 4, 1988, 635 - 641 pp. (in Russian).

[2] M.M. Vainberg, V.A. Trenogin, Branching Theory of Solutions of Nonlinear Equations. Moscow, Nauka Publ., 1969, 527 p. (in Russian).