Page 300 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 9.6 Effect of Rotary Inertia and Shear Deformation 287

rectangular and circular cross sections, the values of k are | and f , respectively. In
addition, there are two dynamical equations:
/ Xr v

(moment) Ji// = — V (9.6-3)

dV
(force) my = + p{x,t) (9.6-4)
where J and m are the rotary inertia and mass of the beam per unit length,
respectively.
Substituting the elastic equations into the dynamical equations, we have

f ^ G \ ^ - «A) -^«A = 0 (9.6-5)
dx \ dx
d
- p(x, t) = 0 (9.6-6)
which are the coupled equations of motion for the beam.
If ijj is eliminated and the cross section remains constant, these two equa­
tions can be reduced to a single equation:
Elm \ d^y
d^y { I Elm Jm d^y
E l + m
dx^ dt 2 \ kAG I dx^ dt^ kAG
(9.6-7)
d^p El d^p
= p{x, t) +
kAG kAG dx^
It is evident then that the Euler equation
d^y
E l —T 4-m — ^ ^ /? (x ,0
dx^ dt^ ^ ^
is a special case of the general beam equation including the rotary inertia and the
shear deformation.

Punge-Kutta method. The Runge-Kutta method is a practical approach
for the structural problem. It is self-starting and results in good accuracy. The
error is of order h^.
To illustrate the procedure, we consider the beam with rotary inertia and
shear terms. The fourth-order equation is first written in terms of four first-order
equations as follows:
dijj M
^ = F { x , ^ , y , M , V )
cfy , V
x
d
dx kAG = G(x, lA, y, M, V)
(9.6-8)
^ = V - = P {x,ik,y,M ,V )
^ = oj^my = K (x,iP,y,M ,V)
dx