Civil Engineering Reference
In-Depth Information
Substitution of the assumed solution into the governing equation yields
i
2
f
sin
2
f
cos
2
−
X
i
f
t
−
Y
i
f
t
+
+
f
(
X
i
cos
f
t
−
Y
i
sin
f
t
)
2
i
2
i
Y
i
cos
f
t
(10.146)
Equating coefficients of sine and cosine terms yields the algebraic equations
+
X
i
sin
f
t
+
f
t
=
F
0
sin
X
i
Y
i
F
0
0
−
f
+
i
2
i
2
f
2
−
f
+
i
=
(10.147)
2
2
i
2
f
−
for determination of the forced amplitudes
X
i
and
Y
i
.
The solutions are
F
0
f
2
i
2
−
X
i
=
f
2
f
i
2
i
−
+
+
(10.148)
F
0
f
i
−
+
Y
i
=
f
2
f
i
2
2
i
2
2
2
−
+
+
To examine the character of the solution represented by Equation 10.145, we
convert the solution to the form
p
i
(
t
)
=
Z
i
sin(
f
t
+
i
)
(10.149)
with
X
i
Y
i
X
i
Y
i
tan
−
1
Z
i
=
+
and
i
=
to obtain
F
0
p
i
(
t
)
=
sin(
f
t
+
i
)
(10.150)
f
2
f
+
i
2
i
−
+
tan
−
1
f
+
i
2
2
−
i
=
(10.151)
i
f
−
Again, the mathematics required to obtain these solutions are algebraically
tedious; however, Equations 10.150 and 10.151 are perfectly general, in that the
equations give the solution for every equation in 10.142, provided the applied
nodal forces are harmonic. Such solutions are easily generated via digital com-
puter software. The actual displacements are then obtained by application of
Equation 10.112, as in the case of undamped systems.
The equivalent viscous damping described in Equation 10.140 is known as
Rayleigh damping
[6] and used very often in structural analysis. It can be shown,
by comparison to a damped single degree-of-freedom system that
+
2
i
=
2
i
i
(10.152)
