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9 Forced Vibration of Continuous Systems
9.16 A missile after being launched is excited longitudinally by a thrust F(t) of its
rocket engine at the end x = 0. Determine the equation for the displacement
u(x, t) and the acceleration ¨
u(x, t).
9.17 A spring of stiffness k is attached to a uniform beam, as shown in the figure.
Show that the one-mode approximation results in the frequency equation.
p
p 1
2
= 1 + 1.5
k
m
M L
3
π 4 E I
where
p
2
1 =
π
4 E I
M L 3
.
Prob. 9.17
9.18 A uniformly simply supported beam is excited by a step moment shown in
the figure. Derive an expression for the elastic bending moment of the beam.
Prob. 9.18
9.19 A uniform rod is connected to a support through a torsional spring of stiffness
k as shown. For the case in which GJ/kL = 1, the expressions for natural
frequencies and mode shapes are
p r = (r − 1) π
G J
I L 2
φ r (x) = cos (r − 1)
π x
L
9 Forced Vibration of Continuous Systems
9.16 A missile after being launched is excited longitudinally by a thrust F(t) of its
rocket engine at the end x = 0. Determine the equation for the displacement
u(x, t) and the acceleration ¨
u(x, t).
9.17 A spring of stiffness k is attached to a uniform beam, as shown in the figure.
Show that the one-mode approximation results in the frequency equation.
p
p 1
2
= 1 + 1.5
k
m
M L
3
π 4 E I
where
p
2
1 =
π
4 E I
M L 3
.
Prob. 9.17
9.18 A uniformly simply supported beam is excited by a step moment shown in
the figure. Derive an expression for the elastic bending moment of the beam.
Prob. 9.18
9.19 A uniform rod is connected to a support through a torsional spring of stiffness
k as shown. For the case in which GJ/kL = 1, the expressions for natural
frequencies and mode shapes are
p r = (r − 1) π
G J
I L 2
φ r (x) = cos (r − 1)
π x
L