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0593_C13_fm Page 443 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 443

Equation (13.2.25) then becomes:

mx cx kx = Fcos pt (13.2.27)
+
+
˙˙
˙
From References 13.9 to 13.13, we see that the general solution may be expressed as:
x = x + x (13.2.28)
h p

where x is the general solution of the homogeneous equation (right side equal to zero)
h
as in Eqs. (13.2.18) and (13.2.19), and where x is any solution of the nonhomogeneous
p
equation (right side equal to Fcospt, as in Eq. (13.2.27)). x is commonly called the particular
p
solution. From Eq. (13.2.19). we see that x is:
h
−µ
x = e [ cos ω t Bsin ω t] (13.2.29)
+
t
A
h
where, as before, µ and ω are defined by Eq. (13.2.20). Also, from the references, we see
that x may be expressed as:
p
x = ( F ) ( [ − 2 + ]
∆
p k mp ) cos pt cpsin pt (13.2.30)
where ∆ is defined as:
−
D 2 2 22
∆ = (kmp ) + c p (13.2.31)
(The validity of Eq. (13.2.26) may be verified by direct substitution into Eq. (13.2.23).)
Finally, if c is zero in Eq. (13.2.27), we have the forced undamped linear oscillator
equation:

mx kx = Fcos pt (13.2.32)
+
˙˙
From Eqs. (13.2.28), (13.2.29), and (13.2.30), we see that the solution is:

x = x + x (13.2.33)
h p
where x and x are:
h
p
x = Acosω t Bsinω t (13.2.34)
+
h
and
x = ( [ F k mp )]
−
2
p cos pt (13.2.35)
where from Eq. (13.2.20) ω is defined as:

D
ω = km (13.2.36)

In the following section, we will examine the undamped linear oscillator in more detail.