1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 305

Case 2. If y' is assumed to be small compa red to 1 (that is, if the bending

of the b eam is small , then (y')^2 is very small compared to 1 and may be

neglected. So equation (37) becomes

( 40)

II -M

y =EI.

Differentiating this equation twice, we obtain

(^4 ) = _!f__ (-M)

y dx^2 EI.

a. If, in addition, I is assumed to b e constant, then substitut ing from
equations (38) we obtain the following simple linear fourth-order differential
equation for the deflection of the b eam

( 41) (^4 ) = -1 (d

2

M) = w(x)

y EI dx^2 EI.

Once the load, w(x), and the initial or boundary conditions a re sp ecified the
deflection of the beam, y(x), at a ny point can easily be calculated by finding
the general solution of ( 41) and satisfying t he initial or boundary conditions.

b. If I is assumed to be a variable, then one differentiation of ( 40) followed
by substitution from (38) and (40) yields

( 42)

MI' M' I' II

y<

3

) = E I 2 - EI= -Iy

v

EI

Differentiation of ( 42) followed by some algebraic rearrangement and substi-
tution from (38) and (41), gives

y(4) = --yI' (3) + (I')- yll - - yll + --- -

2

I" VI' V'

. I I I EI^2 EI

2I' (3) I

11

II w

=--y --y + -

I I EI'

So if we assume y' is small comp ared to 1 and I is a variable, we obtain
the foll owing linear fourth-order differential equation for the deflection of the
beam

(43)