116
SINGLE DEGREE OF FREEDOM STRUCTURES
these models in response to applied environmental loads pi(t). The governing
équation considered now is of the linear form
mv + C]V + k\v = pi(t)
(5.58)
where it is recalled that the subscript (1) on the parameters is the désignation
for a single degree of freedom System. In this section, v(t) is computed for a
harmonie load and an impulsive load. Then the response fonction is derived for
a general loading fonction.
Harmonie Response Fonction
The harmonie response fonction
sometimes called the complex frequency response fonction, is derived from eqoation (5.58) in the following way.
Choose a harmonie loading fonction of constant magnitude po and frequency w,
or
Pi(i)=Po^
(5.59)
Then choose a harmonie solution of équation (5.58) in the form
v(t) = ^//(u’)c^'
(5.60)
fci
After substituting équations (5.59) and (5.60) into équation (5.58), it follows
that
tfmw2 + jC1u + kJe^H^) = kxeiut
(5.61)
Since j2 = -1, the harmonie response function is
H\i = fci(-mu? + jc\w 4- fci)"1
(5.62)
and its modulus is
H(w) ■
- mw2)2 + c2w2]~1/2
(5®)
Note that for a linear structure in which the independent coordinate is the
rotation 9 instead of v, the frequency response function is similar in form to
équation (5.62), but the constants would hâve a different meaning (Jo would
replace m, 9 would replace v and so on).
In some applications it is convenient to recast 7/(cu) and its modulus in
terms of the system’s undamped natural frequency cuq and the damping ratio
Ç, defined by
Cl
2x/kxm
(5.64)
In terms of these latter définitions, the modulus of équation (5.63) becomes
(5.65)