Page 55 - 4560

2
We introduce the notation P EJ  k and we obtain a
кр min
homogeneous linear differential equation
2
d y 2
 k y  0. (5.3)
dx 2
The general integral of this equation is a harmonic function
y  A sin kx  B coskx , (5.4)
where A and B – steel integration, determined by the conditions
fixing the ends of the rod, i.e., with boundary conditions.
Swivel fixing the ends of the rod eliminates variations in its
extreme points, i.e., the boundary conditions are fulfilled: if
0
l
0
x  0 then y  , if x  then y  . Substituting the boundary
conditions in the solution (5.4), we obtain a system of two
equations
0  B ; 0  A sin kl . (5.5)
The constant of integration A must not be zero, otherwise
excluded curvature of the rod. Hence, we conclude that the
second of equations (5.5) to zero is another of the factors:
0
sin kl  ,
where
kl  n , (5.6)
where n – any integer.
Given the previously introduced notation k  P EJ of
кр min
relations (5.6) define a set of critical forces:
2
 2 n EJ
P  min . (5.7)
кр 2
l
In calculating the stability practical importance has the least
critical force corresponding equality n  :
1
 2 EJ
P  min . (5.8)
кр 2
l
Equality (5.8) is called Euler's formula.
Bent axle rod in a state of equilibrium is indifferent sine equation
which
y  A sin  x l  , (5.9)

55