Page 475 - Dynamics of Mechanical Systems

0593_C13_fm Page 456 Monday, May 6, 2002 3:21 PM

456 Dynamics of Mechanical Systems

TABLE 13.8.1
Modes of Vibration of Spring-Supported Particles
Mode Frequency Normalized Amplitudes

2 − ( [ )] 12
/
/
1 2)(km A 1 = 1 2, A 2 = 2 2 , A 3 = 1 2
/
2 2 [ km ] 12 A 1 = 2 2/, A 2 = 0, A 3 = − 2 2
/
2 + ( [ )] 12
/
/
3 2)(km A 1 =− 1 2, A 2 = 2 2, A 3 = − 1 2
Recall that we found not one but three nontrivial solutions to these equations. Each solution
had its own frequency, which means that the system can vibrate in three ways, or in three
“modes,” as depicted in Figures 13.7.4, 13.7.5, and 13.7.6. These are called the natural modes
of vibration of the system.
To discuss this further, consider the amplitudes of these vibration modes as in Eqs.
(13.7.23), (13.7.27), and (13.7.31) and as listed in Table 13.8.1. In view of the amplitude
ratios, let new variables ξ , ξ , and ξ be introduced as:
2
1
3
ξ = ( 1 ( 2 (
+
+
D
1 12)x 2 2) x 12)x 3 (13.8.4)
ξ = ( + 2 (
−
D
2 22) x 1 0x 2 2) x 3 (13.8.5)
ξ = −( 1 ( 2 (
−
+
D
3 12)x 2 2) x 12)x 3 (13.8.6)
Then, by differentiating, we have:
ξ = −( ˙˙ +( ˙˙ +( 12) ˙˙ x
˙˙
D
1 12)x 1 2 2) x 2 3 (13.8.7)
ξ = ( ˙˙ −( 22) ˙˙ x
˙˙
D
2 22) x 1 3 (13.8.8)
ξ = −( ˙˙ +( ˙˙ +( 12) ˙˙ x
˙˙
D
3 12)x 1 2 2) x 2 3 (13.8.9)
˙˙
Consider first ξ 1 . By substituting from Eqs. (13.8.1), (13.8.2), and (13.8.3), we have:
)(
ξ = ( 12)(km ) − x + x 2) +( 2 2)(km x − 2x + x 3)
2 (
˙˙
1 1 1 2
)(
+( 12)(km x 2 − 2x 3)
2 ( [
+ (
+
2 2 x
= (km ) −+ 2 2)] x 1 ( [ 1− 2 x 2 )] [ 2 − ) 3 ]
[
2 (
)
+
12
= (km ) −+ 2) ( 1 2)x 1 ( 2 2) x 2 +( )x 3 ]
2 ξ
( 2
= (km ) −+ ) 1