36
2 Single Degree of Freedom (SDOF) Systems
{1, cos(ν 1 t), sin(ν 1 t), · · · , cos(ν i t), sin(ν i t), · · · }
and that the Fourier coefficients are projections of f (t) on these basis functions (see
Appendix C). The Fourier series of f (t) is conceptually similar to the representation
of vectors v in, e.g., the three-dimensional Euclidian space, which has the form
v = v x i + v y j + v z k, where i, j, k denote the unit vectors along the x, y, zcoordinates and v x , v y , v z are the projections of v on these unit vectors. Note that
the representation of v and f (t) have the same form. They are sums of projections
of these elements on the orthogonal systems spanning their spaces. Note also that
approximations of f (t) by its values at N < ∞ times in [0, τ ] are vectors in Ndimensional Euclidian spaces.
2.5.2 Steady-State Solution
Consider the oscillator in Eq. 2.8 and approximate the forcing function f (t) by its
truncated Fourier series f n (t) of Eq. 2.56 so that the oscillator displacement x(t) is
approximated by the solution x n (t) of the differential equation
¨
x n (t) + 2 ζ ω ˙
x n (t) + ω
2 x n (t) =
1
m
a 0
2
+
n
i=1
a i cos(ν i t) + b i sin(ν i t)
.
(2.58)
The initial conditions are not specified since we are interested in the steady-state
solution of this equation.
Since the system is linear, x n (t) can be constructed by adding the solutions
of the above equation to the forcing functions a 0 /(2 m), {(a i /m) cos(ν i t)}, and
{(b i /m) sin(ν i t)}. This statement also follows from the Duhamel integral of
Eq. 2.30 which gives
x n (t) =
t
0
h(t − u) f n (u) du =
1
m
a 0
2
t
0
h(t − u) du
+
n
i=1
a i
t
0
h(t − u) cos(ν i u) du + b i
t
0
h(t − u) sin(ν i u) du
since the integral is a linear operator.
We have already calculated the steady-state solutions to the above forcing
functions, see Eq. 2.14, Eq. 2.48 and Eq. 2.51. These input-output relationships
applied to the terms in the representation of f n (t)/m are
a 0 /(2 m) ⇒ a 0 /(2 m ω
2 )
(a i /m) cos(ν i t) ⇒ (a i /m)
r d (ν i )/ω
2
cos(ν i t − ϕ i )
2 Single Degree of Freedom (SDOF) Systems
{1, cos(ν 1 t), sin(ν 1 t), · · · , cos(ν i t), sin(ν i t), · · · }
and that the Fourier coefficients are projections of f (t) on these basis functions (see
Appendix C). The Fourier series of f (t) is conceptually similar to the representation
of vectors v in, e.g., the three-dimensional Euclidian space, which has the form
v = v x i + v y j + v z k, where i, j, k denote the unit vectors along the x, y, zcoordinates and v x , v y , v z are the projections of v on these unit vectors. Note that
the representation of v and f (t) have the same form. They are sums of projections
of these elements on the orthogonal systems spanning their spaces. Note also that
approximations of f (t) by its values at N < ∞ times in [0, τ ] are vectors in Ndimensional Euclidian spaces.
2.5.2 Steady-State Solution
Consider the oscillator in Eq. 2.8 and approximate the forcing function f (t) by its
truncated Fourier series f n (t) of Eq. 2.56 so that the oscillator displacement x(t) is
approximated by the solution x n (t) of the differential equation
¨
x n (t) + 2 ζ ω ˙
x n (t) + ω
2 x n (t) =
1
m
a 0
2
+
n
i=1
a i cos(ν i t) + b i sin(ν i t)
.
(2.58)
The initial conditions are not specified since we are interested in the steady-state
solution of this equation.
Since the system is linear, x n (t) can be constructed by adding the solutions
of the above equation to the forcing functions a 0 /(2 m), {(a i /m) cos(ν i t)}, and
{(b i /m) sin(ν i t)}. This statement also follows from the Duhamel integral of
Eq. 2.30 which gives
x n (t) =
t
0
h(t − u) f n (u) du =
1
m
a 0
2
t
0
h(t − u) du
+
n
i=1
a i
t
0
h(t − u) cos(ν i u) du + b i
t
0
h(t − u) sin(ν i u) du
since the integral is a linear operator.
We have already calculated the steady-state solutions to the above forcing
functions, see Eq. 2.14, Eq. 2.48 and Eq. 2.51. These input-output relationships
applied to the terms in the representation of f n (t)/m are
a 0 /(2 m) ⇒ a 0 /(2 m ω
2 )
(a i /m) cos(ν i t) ⇒ (a i /m)
r d (ν i )/ω
2
cos(ν i t − ϕ i )