An approach to the response Analysis of Shafts – Discussion

16 02 2010

4 Discussion

The study of shaft behaviour under dynamic loading and rotation is of profound importance in predicting resonance, in system control and system monitoring. The purpose of the work presented in this paper is to introduce the method of Proper Orthogonal Decomposition (POD) as a tool that can be effectively used in characterising the dynamics involved in the above tasks and extracting useful information for the real-time behaviour of the structure. The method is used in the form of meta-processing of finite element simulation results. It has been shown that the POD method is used to reproduce modes and corresponding frequencies that are systematically correlated under variations of initial conditions in dynamical problems of free vibration. Even when combined loading is applied, the POD method correctly discriminates and classifies these modes and frequencies. It has also been shown that the frequencies are affected by dynamical effects and pre-strain, a behaviour that is expected but is often difficult to calculate.The Proper Orthogonal Decomposition presents a considerable advantage: it is indifferent to the system that generates the input to the method. Nevertheless, it succeeds in extracting dominant modes and classifying them from the time-space response of the structure. Throughout this text, POMs have not been considered to coincide necessarily with natural modes of vibration. Rather, POMs are appropriate combinations of modes and therefore feasible configurations of a body. POMs, being orthogonal, they form a basis of the space where the configurations of the body in the particular process lie. Moreover, they are classified in an eigenvalue sense. These two properties are very important. In combination, they identify dominant POD modes in the response. This information can be interpreted in two ways, especially  in cases where a unilateral behaviour is desired. The first interpretation is that a strong dominant mode depicts a process that is consistent and “robust” to that mode. The second interpretation is that singular value percentage dispersion over more than one mode signifies a process that includes strong interference with the dominant mode.

This work has further shown that field participation in each mode is a tool to identify symmetric modes and further describe the content of each mode. Finally, the frequency analysis of the amplitude curves provides information on the response spectrum of the modes. This set of tools can be effectively employed in parametric design, control and monitoring processes. Such an analysis provides qualitative and quantitative evidence of the consistence to a functional behaviour that may be desirable or undesirable. In a design context, this evidence can be used as a criterion of process consistency under parameter variation. In a control and monitoring context, it can be used to identify factors that introduce noise to the process and distract it from the desired functional behaviour.

5 References

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Appendix:Tables and figures

Figure 1: The shaft finite element model and coordinate system orientation.

Figure 2: longitudinal translation of the end face

Figure 3: rotation of the end face around the longitudinal axis

Mode 1

Mode 2

POD U1

POD U2

POD U3

Mode 3

Mode 4

POD U1

POD U2

POD U3

Mode 5

Mode 6

POD U1

POD U2

POD U3

Figure 4: The first six modes for the combined torsion-compression model presented in Paragraph 3.1

Figure 5: Amplitude curves for the first six POMs.

Axial (U1)

Torsional (U2)

Mode 1

Mode 2

Mode 3

Mode 4

Figure 6: U1 fields of four pure axial modes and U2 fields of four pure torsional modes.

Combination (U1)

Combination (U2)

Mode 1

Mode 2

Mode 3

Mode 4

Figure 7: U1 and U2 displacement fields of four modes for combined vibration.

POD U1

POD U2

POD U3

Figure 8: Two first mode shapes for rotating cylinder at 1500 rad/s.

POD U1

POD U2

POD U3

Figure 9: Third mode shape for rotating cylinder at 1500 rad/s.

Energy Distribution %:

U1

U2

U3

mode 1

98.6904

0.998077

0.043939

0.04394

mode 2

1.0901

0.976873

0.151224

0.121226

mode 3

0.1536

0.863442

0.356699

0.35672

mode 4

0.0295

0.739299

0.476154

0.476158

mode 5

0.0249

0.25541

0.683673

0.683657

mode 6

6.37E-03

0.630065

0.549105

0.549111

Table 1: singular value percentage distribution and field participation in each mode.

Natural

Frequencies [Hz]

Natural Frequencies

pre-strain [Hz]

800.33

800.33

1180.5

1180.5

1180.5

1180.5

1307.9

2399.7

2399.7

2439.8

2439.8

2439.8

2439.8

2627.5

Table 2: Natural frequencies of vibration for unstrained and pre-strained model.

Axial displacement (U1) [Hz]

Torsion [Hz]

1312.5

105.47

2132.8

808.59

117.19

1628.9

1382.8

Table 3: Excited frequencies in axial and rotational response.

mode 1

1312.5

mode 2

2132.81

mode 3

105.469

mode 4

1628.91

mode 5

93.75, 808.594

mode 6

2625

Table 4: Frequencies extracted from the amplitude curves.

Displacement

[m]

Rotation [deg]

Singular value percentage

[%]

Frequency [Hz]

POM field

Axial 1

0.001

94.30%

1312.5

U1

Axial 2

0.01

94.31%

1312.5

U1

Axial 3

0.02

94.40%

1300.8

U1

Axial 4

0.05

96.45%

1289.1

U1

Rotation 1

5

73.83%

808.59

U2-U3

Rotation 2

10

65.51%

832.30

U2-U3

Rotation 3

15

57.39%

843.75

U2-U3

Rotation 4

25

61.38%

878.91

U2-U3

Combination 1

0.001

5

73.45%

808.59

U2-U3

Combination 2

0.01

10

58.83%

832.03

U2-U3

Combination 3

0.02

15

51.53%

843.75

U2-U3

Combination 4

0.05

25

51.77%

1335.9

U1

Table 5: Initial conditions for the free vibration load cases, singular value percentage of dominant mode, dominant frequencies and related POM fields.

Load Case POM Fq.1 Fq.2 Fq.3 Fq.4 Fq.5 Singular value percentage U1 U2 U3
Axial 1 1 1312.5 94.30% 0.9981 0.04418 0.0442
2 2144.5 4.99% 0.9778 0.14832 0.1483
3 82.031 0.64% 0.9001 0.3079 0.3079
4 1570.3 0.06% 0.7412 0.4747 0.4747
Axial 2 1 1312.5 94.31% 0.9981 0.04383 0.0438
2 2144.5 4.97% 0.9778 0.14813 0.1481
3 82.031 0.63% 0.9017 0.30575 0.3058
4 1570.3 1699 1383 0.06% 0.7416 0.47437 0.4744
Axial 3 1 1300.8 94.40% 0.9982 0.04295 0.043
2 2156.2 4.90% 0.9783 0.14645 0.1465
3 70.312 0.60% 0.9038 0.30264 0.3026
4 1582 1688 1383 0.07% 0.742 0.47402 0.474
Axial 4 1 1289.1 96.45% 0.9983 0.04174 0.0417
2 2179.7 2520 890.6 2.92% 0.9788 0.14491 0.1449
3 70.312 11.72 1231 375 1359 0.28% 0.9071 0.29765 0.2977
4 1664.1 1359 1594 0.19% 0.731 0.48247 0.4825
Torsional 1 1 808.59 73.83% 0.0039 0.7071 0.7071
2 23.438 58.59 16.85% 0.9997 0.017 0.017
3 2437.5 6.80% 0.0088 0.70708 0.7071
4 1945.3 1.73% 0.0032 0.70711 0.7071
Torsional 2 1 832.3 65.51% 0.0103 0.70707 0.7071
2 23.438 1324 58.59 26.08% 0.9975 0.0504 0.0504
3 2519.5 6.07% 0.0018 0.70712 0.7071
4 1828.1 2520 1.43% 0.003 0.70712 0.7071
Torsional 3 1 843.75 57.39% 0.042 0.70648 0.7065
2 23.438 1336 843.8 35.35% 0.9938 0.07885 0.0788
3 2566.4 4.99% 0.0153 0.70703 0.707
4 1710.9 1.39% 0.0144 0.70704 0.707
Torsional 4 1 878.91 61.38% 0.0396 0.70666 0.7064
2 11.719 1359 878.9 1769.5 30.04% 0.9938 0.07881 0.0782
3 2683.6 2625 2578 2519.5 5.52% 0.0092 0.7072 0.707
4 1582 1652 1781 1699.2 1.13% 0.0316 0.70681 0.7067
Combination 1 1 808.59 73.45% 0.0044 0.7071 0.7071
2 23.438 1313 58.59 1617.2 17.22% 0.9997 0.01649 0.0165
3 2437.5 6.77% 0.0089 0.70708 0.7071
4 1945.3 1.72% 0.0027 0.70711 0.7071
Combination 2 1 832.03 58.83% 0.019 0.70698 0.707
2 1324.2 23.44 58.59 820.31 32.57% 0.9994 0.02414 0.0243
3 2519.5 5.48% 0.0065 0.7071 0.7071
4 1828.1 1.30% 0.0299 0.7068 0.7068
Combination 3 1 843.75 51.53% 0.0571 0.70595 0.706
2 1324.2 23.44 843.8 1675.8 40.45% 0.9976 0.04905 0.0491
3 2543 4.32% 0.0163 0.70702 0.707
4 1757.8 1324 2074 23.438 1.29% 0.1224 0.7018 0.7018
Combination 4 1 1335.9 11.72 855.5 93.75 51.77% 0.9917 0.09084 0.0912
2 855.47 1336 23.44 480.47 40.92% 0.117 0.7023 0.7022
3 2613.3 2555 2520 2484.4 3.13% 0.0616 0.70588 0.7057
4 2015.6 23.44 2074 2109.38 1336 1.84% 0.9678 0.17885 0.1773

Table 6: Free vibration POD results for different initial conditions.

Energy Distribution:

U1

U2

U3

Mode 1

49.8523%

0.001112

0.707084

0.707128

Mode 2

49.5715%

0.000745

0.70713

0.707083

Mode 3

0.2644%

0.030459

0.706808

0.70675

Table 7: Singular value percentage allocation and POM Field participation, 1500 rad/sec.

Energy Distribution:

U1

U2

U3

Mode 1

55.2160%

2.89E-06

0.707082

0.707132

Mode 2

44.7820%

4.32E-05

0.707132

0.707082

Mode 3

0.0011%

0.000648

0.707132

0.707082

Table 8: Singular value percentage allocation and POM Field participation, 150 rad/sec.

Energy Distribution:

U1

U2

U3

Mode 1

99.0559%

2.34E-05

0.707125

0.707088

Mode 2

0.9425%

0.000134

0.707088

0.707125

Mode 3

0.0011%

0.992436

0.086807

0.86811

Table 9: Singular value percentage allocation and POM Field participation, 15 rad/sec.

Energy Distribution:

U1

U2

U3

Mode 1

99.8701%

0.000256

0.70711

0.707104

Mode 2

0.1091%

0.99208

0.088816

0.088817

Mode 3

0.0099%

0.689706

0.512006

0.512011

Table 10: Singular value percentage allocation and POM Field participation, 1.5 rad/sec.

Energy Distribution:

U1

U2

U3

Mode 1

54.3742%

0.000295

0.707091

0.707122

Mode 2

45.6253%

0.000240

0.707122

0.707091

Table 11: Singular value percentage allocation and POM Field participation, 0-1500 rad/sec.

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