Page 205 - Handbook of Materials Failure Analysis

5 Case Study 201

Table 8.7 Evaluation of the Stiffness Matrix Elements
134,915.35 N/m
K 00 ¼K 1 ¼
134,915.35 N/m
K 01 ¼K 10 ¼ K 1 ¼
303,131,565.91 N/m
K 11 ¼K 0 +K 1 ¼

K 00 K 01 2 M 0
0
¼ 0 (8.17)
ω
K 10 K 11 0 M 1

2
K 00 ω M 0 K 01

¼ 0 (8.18)
2
K 10 K 11 ω M 1

The determinant will be:
2 2
K 00 ω M 0 K 11 ω M 1 K 01 K 10 ¼ 0 (8.19)
The resolution of this equation gives us the vibration pulsations of the two main
modes:
2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
s
1 K 00 K 11 K 00 K 11 2 K 01 K 10
ω 2 4 + + +4 5 (8.20)
1,2
2 M 0 M 1 M 0 M 1 M 0 M 1
¼
Knowing the pulsations of the eigenmodes ω 1 and ω 2 , we deduct the periods of the
two main modes:
2π 2π
T 1 ¼ and T 2 ¼
ω 1 ω 2
The calculation of the pulsations and periods of the two main eigenmodes is given in
the Table 8.8.
5.3.5 Determination of the eigenmodes
The system has a same number of degrees of freedom than eigenmodes. For the

a i0
i
determination of the eigenmodes a ¼ , we must solve the equation:
fg
a i1
2
K ½Š ω M½Š a ¼ 0 (8.21)
i
i
fg
fg
Table 8.8 Evaluation of the Pulsations and Periods of the Two Main
Eigenmodes
ω 2 Pulsation ω Period T (s)
Mode 1 5.95 2.44 2.58
Mode 2 2,385.63 48.84 0.13