PSI - Issue 22

E. Georgievskaia / Procedia Structural Integrity 22 (2019) 229–234 E. Georgievskaia / Structural Integrity Procedia 00 (2019) 000 – 000

232

5. The calculation results The following dynamic parameters characterizing the dynamic system response have been defined: the eigenvalues and corresponding mode shapes, participation factors, effective masses for each direction of excitation for each eigenvalue. Some calculation results for the first four eigenvalues presented in Table 1 and on Fig. 3-5. Participation factors and effective masses characterize the reaction of the support under external influences. If they are equal to zero, it is impossible to determine the magnitude of the stress in runner blades by the support vibration values. For example, participation factors and effective masses at mode 4 in all directions (see Table 1) are equal to zero. At the same time, dynamic stresses in the runner blades act (see Fig. 4) and they may cause cracking in the zone of joint of the blades with the hub or rim. The Fig. 3-5 present mode shapes, amplitudes of dynamic stress intensity and displacements corresponding to the modes under consideration. All values are given in normalized forms (the maximum value is 1.0). The pie charts show these values for the 16 design points equivalent to the 16 blades in area of the output edge intersection with the runner. Fig. 6 shows the eigenvalue changes depending on the length of the growth through model crack L. Fig. 6(a) presents the calculation results for a single blade in relative terms (the crack length – as a percentage of the total length of the interface line of the blade with the hub, the frequency – as a percentage of the corresponding eigenvalue changes by comparison to the blade without a crack). Fig. 6(b) demonstrates results for the whole runner: without crack, with long crack (length – 25%) on the one blade, with identical cracks the same length on all blades.

Table 1. Dynamic parameters. Mode Eigenvalues, Hz

Participation factors

Effective masses

Х

Y

Z

R x

R y

R z

Х

Y

Z

R x

R y

R z

1 2 3 4

16,74 16,74 22,29 28,89

149

80,7 149

0 0

130 188

-188

0 0

0,281 0,083 0,083 0,281

0 0

0,056 0,115 0,115 0,056

0 0

-80,7

130

0 0

0 0

12,7

0 0

0 0

-522

0 0

0 0

0,002

0 0

0 0

0,966

0

0

0

0

Notice: X, Y, Z, Rx, Ry, Rz – the direction of movement and rotation relative to the coordinate axes

Fig. 3. Mode shapes: (a) skew-symmetric bending mode; (b) torsional mode; (c) symmetric bending mode.