Civil Engineering Reference
In-Depth Information
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Advanced Engineering Dynamics,
2nd ed. New York: Cambridge
University Press, 1985.
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Classical Mechanics,
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Finite Element Procedures.
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13.
PROBLEMS
10.1
Verify by direct substitution that Equation 10.5 is the general solution of
Equation 10.4.
10.2
A simple harmonic oscillator has
m
=
3 kg,
k
=
5 N/mm
.
The mass receives
an impact such that the initial velocity is 5 mm/sec and the initial displacement
is zero. Calculate the ensuing free vibration.
10.3
The equilibrium deflection of a spring-mass system as in Figure 10.1 is
measured to be 1.4 in. Calculate the natural circular frequency, the cyclic
frequency, and period of free vibrations.
10.4
Show that the forced amplitude given by Equation 10.28 can be expressed as
X
0
1
−
r
2
U
=
r
=
1
with
X
0
=
F
0
/
k
equivalent static deflection
and
r
=
f
/
≡
frequency ratio.
10.5
Determine the solution to Equation 10.26 for the case
f
=
.
Note that, for
this condition, Equation 10.29 is
not
the correct solution.
10.6
Combine Equations 10.5 and 10.29 to obtain the complete response of a simple
harmonic oscillator, including both free and forced vibration terms. Show that,
for initial conditions given by
x
(
t
=
0)
=
x
0
and
x
(
t
=
0)
=
v
0
, the complete
response becomes
X
0
1
−
r
2
(sin
f
t
−
r
sin
t
)
with
X
0
and
r
as defined in Problem 10.4.
v
0
x
(
t
)
=
sin
t
+
x
0
cos
t
+
10.7
Use the result of Problem 10.6 with
x
0
=
v
0
=
0,
r
=
0
.
95,
X
0
=
2,
f
=
10 rad/sec
and plot the complete response
x
(
t
) for several motion cycles.
