Page 363 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

350 Mode-Summation Procedures for Continuous Systems Chap. 11

which has the solution

^ -y/llwo (-1 )"
Qnit) mrMQ (1 - cos oj^t) 0 < t < t]

y/2lw„ ( - 1 ) ”
------coso>„0
mrM, 0
2v/2K( -1 )" r, < r < 00
iivMqw^ [1 - 'l)]
Thus, the deflection of the beam is expressed by the summation
Trnx
) = I
Example 11.1-2
A missile in flight is excited longitudinally by the thrust F{t) of its rocket engine at
the end jr = 0. Determine the equation for the displacement u(x, t) and the accelera­
tion ii(x, t).

Solution: We assume the solution for the displacement to be
u{x,t) =
where (p^ix) are normal modes of the missile in longitudinal oscillation. The general­
ized coordinate satisfies the differential equation
F(/)y,(0)

If, instead of F(t), a unit impulse acted at x = 0, the preceding equation would have
the solution [(p-(0)/M,wJsin 0)^1 for initial conditions (3^-(0) = q(0) = 0. Thus, the
response to the arbitrary force F(t) is
y,(o)
^ l é ^ / ' F ( f ) s i n a.,(i
and the displacement at any point x is

u(x,t) = L f'F ii)sin (0;(t - i ) d i
M:0)i
The acceleration ¿¡¿{t) of mode i can be determined by rewriting the differential
equation and substituting the former solution for qi(t):

di{t) = - <o?<J,(0

^(O<p,(0) 'i>,(o)io, fi
('F(i)sin io,(i -
M, M, i ■'0 dn