Page 399 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

386 Classical Methods Chap. 12

Figure 12.3-1.
Solution: For the displacement function, we choose the first two longitudinal modes of a
uniform rod clamped at one end.
. . . 7TX . 3 ttx
u{x) = Cj sm + C2 sm 2/
/
= Cj(/)j(x) + C2<>2(-^) (a)
The mass per unit length and the stiffness at x are
m{x) = - y j and EA(x) = Ea J^I - y j

The and the for the longitudinal modes are calculated from the equations
k , j = j ' ^ E A { x ) 4 > ] 4 > ] d x

mi,= j ' m { x ) < ! > j 4 > j d x

" 4/ 0 2 1 ^ 2/ ( 8 2 )
EA,
= 0.86685
I
X \ 77X 3tTX , EA,
kn = ^ E A 4 ' - î COS -7^ cos ~Yj- dx = 0.750 I (b)
4/2
'2
23 ttx . _ ^-^0 / 97t^ J_\

= - ^ E A cos 2/ dx 2/ \ 8 ^ 2 j
4r 0
E^{)
= 5.80165
mil '^0/ ~ 7 ) ^ ^ ----^ j 0.148679mQ/
3 tTX ,
TTX .
X \ .
mi2 = miij - y j sm -yy sm 2/ ^ ,/ 1 \ ^
—2 I ^ 0.101321 mo/
n iji = ^ { ) j (^ “ 7 ) ~ 7 ~ ^ ^ ~ j ^ 0.238742mo/
Substituting into Eq. (12.3-7), we obtain

|o .8 6 6 8 5 ^ - 0.14868wo/t^^j |o . 7 5 0 ^ - 0.10132/n„/i^^)

(o .7 5 0 -^ - 0.10132m„/o>^j (5 .8 0 1 6 5 ^ - 0.23874mo/a)^j
(c)