Page 101 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

98 Chapter 1 Fundamentals oF Vibration
either a time domain or a frequency domain. Figure 1.57 shows that the frequency-domain
representation does not provide the initial conditions. However, in many practical appli-
cations the initial conditions are often considered unnecessary and only the steady-state
conditions are of main interest.

1.11.5 An even function satisfies the relation
even and odd x1-t2 = x1t2 (1.87)
Functions
In this case, the Fourier series expansion of x(t) contains only cosine terms:
a
x1t2 = 0 + a cos nvt (1.88)
a n
2 n = 1
where a and a are given by Eqs. (1.71) and (1.72), respectively. An odd function satisfies
n
0
the relation
x1-t2 = - x1t2 (1.89)
In this case, the Fourier series expansion of x(t) contains only sine terms:

x1t2 = a n (1.90)
b sin nvt
n = 1
where b is given by Eq. (1.73). In some cases, a given function may be considered as even
n
or odd depending on the location of the coordinate axes. For example, the shifting of the
vertical axis from (a) to (b) or (c) in Fig. 1.58(i) will make it an odd or even function. This
means that we need to compute only the coefficients b or a . Similarly, a shift in the time
n
n
axis from (d) to (e) amounts to adding a constant equal to the amount of shift. In the case of
Fig. 1.58(ii), when the function is considered as an odd function, the Fourier series expan-
sion becomes (see Problem 1.107):
4A 1 2p12n - 12t
x 1t2 = a sin (1.91)
1
p n = 1 12n - 12 t
On the other hand, if the function is considered an even function, as shown in Fig. 1.50(iii),
its Fourier series expansion becomes (see Problem 1.107):
4A 1-12 n + 1 2p12n - 12t
x 2 1t2 = a cos (1.92)
p n = 1 12n - 12 t

Since the functions x 1t2 and x 1t2 represent the same wave, except for the location of the
2
1
origin, there exists a relationship between their Fourier series expansions also. Noting that
t
x ¢t + ≤ = x 1t2 (1.93)
1
4 2