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 realtime behaviour of the structure. The method is used in the form of metaprocessing 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 prestrain, 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 timespace 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.
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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 torsioncompression 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 
6.37E03 
0.630065 
0.549105 
0.549111 
Table 1: singular value percentage distribution and field participation in each mode.
Natural Frequencies [Hz] 
Natural Frequencies prestrain [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 prestrained 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 
U2U3 
Rotation 2 

10 
65.51% 
832.30 
U2U3 
Rotation 3 

15 
57.39% 
843.75 
U2U3 
Rotation 4 

25 
61.38% 
878.91 
U2U3 
Combination 1 
0.001 
5 
73.45% 
808.59 
U2U3 
Combination 2 
0.01 
10 
58.83% 
832.03 
U2U3 
Combination 3 
0.02 
15 
51.53% 
843.75 
U2U3 
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% 
2.89E06 
0.707082 
0.707132 
Mode 2 
44.7820% 
4.32E05 
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% 
2.34E05 
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, 01500 rad/sec.
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