# 3Numerical examples

In the following examples, finite element simulations are used to demonstrate this concept. These simulations concern shaft loading with torsional moment and axial compression. This is a common loading condition for shafts, for example in rotating machinery used in power plants. For performing the simulations, the commercial finite element code Abaqus/Explicit 6 has been employed, using double precision arithmetic. Further, the POD processing is performed with custom software based on the LAPACK library 6. In all cases, POD analysis is performed over at least 300 snapshots (taking care to always observe the lower bound of the Nyquist–Shannon sampling theorem) including all the degrees of freedom of all nodes in the models. The results presented hereafter include the extracted mode shapes and plots of variable field distributions, amplitude vs. time fluctuation for each mode and Singular value percentage for each mode. It is noted here that in all cases multi-field POD is performed. This means that the input snapshot vectors include three displacement variables for each node. The resulting POMs include fields that correspond to each of the input variables. The amplitude vs. time curves for each mode represents the variation with time of the mode participation in the time-space domain of the simulation. In vibration problems this is usually an oscillating curve and a Fourier analysis is performed on this time variation so as to calculate the excited frequencies for each mode 6. Finally, the singular values relative percentage presents an overall estimated participation factor for the mode. Considering the fact that in the results there are three fields to each POM, it is interesting to calculate the norm of each field so as to determine its participation to the POM. In the results presented, the sum of the squared field norms equals unity and the field norms are presented next to the singular value percentage so as to easily determine which field is dominating the POM.

## 3.1Shaft response to torsional vibration and axial compression

A very common loading condition for shafts is the simultaneous application of a torsional moment and a compressive axial force. The corresponding simulations have been used as baseline for comparison and mesh convergence for the subsequent simulations. For this purpose, thorough investigation of the results has been performed. The simulations presented involve short cylindrical finite element models. The height of the cylinder is 1 m and the diameter is 0.4 m. Multi-point constraints are imposed to the two bases of the cylinder so that these behave as flat disks. The model used in this set of simulations features a quite dense finite element mesh: 101101 nodes and 96000 elements are employed. The longitudinal axis of the shaft is aligned with the x-axis of the coordinate system. The two bases of the shaft lay on planes parallel to the y-z plane (Figure 1). The shaft is considered to be made of steel. No material nonlinearities are considered. On the other hand, geometric nonlinearities are included in all analyses.

A torsional moment and an axial force are applied to the shaft. The moment of 100 kNm is applied to the master node of the Multi-point constraint at a frequency of 100 Hz. The compression load is applied at the same node and is equal to 100 kN. The central node at the loaded face translates (Figure 2) and rotates along the longitudinal axis (Figure 3).

In this context, the first six POMs are presented herein (Figure 4). The corresponding amplitude curves are presented in Figure 5. Table 1 presents the energy distribution between POMs and the participation of the corresponding POM displacement fields to each mode. It is straightforward that most of the Singular value percentage is accumulated in the axial displacement of the first mode and second mode. Also, it is notable that there is an almost symmetric distribution of energy between the fields that correspond to the transverse displacement variables. The first mode represents a first order compression-tension mode and the second mode a second order compression-tension mode since there is a stationary section, as represented by the green area at the mid-span of the shaft, Figure 4.

It is interesting to investigate the way vibration frequencies are classified by the POD method. Table 3 includes the frequencies that are excited in the axial and rotational response of the loaded base of the shaft. Table 2 presents the extracted natural frequencies from eigenvalue analyses. Two sets of natural frequencies are presented: the first column of Table 2 includes the natural frequencies of the unloaded model. The second column includes the natural frequencies when the axial load of 100 kN is used to pre-strain the model. As expected, pre-straining the model introduces shifts in the natural frequencies. It is notable that there is excitation of frequencies that do not correspond to torsional vibration only, due by the coupling of the two loading conditions. Table 4 provides the classification of the frequencies extracted from the amplitude curves (Figure 5). The POM classification provides also a classification of the frequencies that are excited in the particular simulation. These frequencies are affected by dynamic effects, nonlinear behaviour and pre-straining. Thus, the POMs, amplitude curves and frequency extraction, and singular value percentages, provide a precise identification of the dynamics of the shaft, including all effects that it is possible to reproduce with the particular simulation.

## 3.2Free vibration of shaft

Further to the detailed identification of the dynamics of the shaft, as presented in Paragraph 3.1, it is important to investigate the POD modes and frequencies for free vibration of the shaft, as well as the effect of load magnitude and load combination. Towards this, a large series of finite element simulations is performed. These simulations include pure axial and pure torsional initial excitation of cylindrical shafts as well as combination of the two. Three load levels and four modes from each case are presented as illustrative examples of the analysis. A finite element model, coarser than the previous one, is used in the simulations so as to reduce run-times. The coarser mesh has been selected so as to adequately represent at least the four first modes of the dense baseline model. The finite element simulations include two phases: in the first one, a displacement is applied slowly and gradually to the master node of the multi-point constraint up to the desired level. Then, the node is released and the shaft is allowed to freely vibrate. The response is captured with an explicit time integration scheme. The initial displacement is an axial displacement, a rotation or a combination of the two. The initial conditions for the simulations presented herein are included in Table 5.

Detailed results of the simulations can be found in the Appendix. For convenience, some results are included in Table 5. For each case, the dominant mode is examined. For this mode, the corresponding frequency as calculated from the amplitude curves as well as the POM field that participates more in the mode are reported. Considering the behaviour to axial excitation, it is straightforward that as the initial displacement increases, the singular value percentage accumulated to the first mode also increases. As expected, the POM field that participates the most in this mode is the field that corresponds to the axial displacement. On the other hand, it is noteworthy that the dominant frequency in the amplitude response decreases as the initial displacement increases. A closer look at the results in Table 6 shows that as the initial displacement increases, more frequencies become distinct. The shift in dominant frequencies shows, as in the case of the forced vibration, that for such problems where loads give rise to geometric nonlinearities, even if the strains are yet small and the mass is evenly distributed, the load level affects the response spectrum.

In the cases where a rotation is initially applied to the end face of the shaft, it is seen that there is not a proportionate correlation between the initial rotation angle and the singular value percentage. On the other hand, the dominant frequency shifts to larger values as the rotation angle increases. The rotation of the end face gives rise to more complex deformation modes than the axial load cases. Barrel and axial distortions are also evident along with the torsional distortion. This is evident in the distribution of the singular value percentage: the first mode allocates considerably smaller amounts of energy in this set of loading conditions than in the set of axial loading conditions. Examination of Table 6 reveals that the second mode that corresponds to axial vibration, also allocates a considerable percentage of energy.

The dependence on the level of initial displacements is even more evident when the two sets of initial conditions are combined. In these simulations, the correlation between singular value percentage and load level resembles the previous one. Only in this case, it is seen that for the load case with maximum initial displacements (i.e. Combination 4 in Table 5) the dominant POM field corresponds to axial displacement instead of transverse displacements. This is also projected on to the dominant frequency, which is shifted towards a value that corresponds to the values of axial loading.

These simulations reflect the fact that the response of continua can be quite complicated even under simple loading conditions. The POD method discriminates between the different modes of deformation and efficiently classifies POMs, based on quantitative evidence of their participation in the recorded response. Moreover, POMs are related to particular frequencies extracted from the related amplitude curve. These frequencies are affected by the loading condition due to effects introduced by dynamics and pre-straining. Their relation to POMs appears to be quite systematic.

## 3.3Rotating shafts

The previous examples demonstrated the response of shafts under axial and torsional vibration. In these cases, dynamic effects are introduced through oscillation. Another considerable aspect in the response of shafts is the introduction of dynamic effects due to rotation. In this context, only the identification of rigid body rotation is included for illustration purposes. In the following simulations, a rotating shaft is considered. The loading and constraint setup is the same as in Paragraph 3.1. The simulation is performed in two steps. The shaft is linearly accelerated to the desired rotational velocity. Then, an alternating torsional moment is applied. In addition, the cylinder is loaded with a compressive load. Results for four rotational velocities ranging between 0 and 1500 rad/sec are presented. A separate case is presented for accelerating the cylinder from 0 to 1500 rad/sec with simultaneous compression and torsional vibration. The POD analysis is performed after the cylinder is accelerated to the desired rotational velocity level. Figure 8 and Figure 9 present the first three POMs for the rotational velocity of 1500 rad/sec. Table 7 to Table 11 present the singular value percentage distribution and POM field participation in each mode. It is noteworthy that for high rotation velocities the vast amount of singular value percentage is equally shared between the first two modes that correspond to the rigid body rotation. The fact that the first modes correspond to rigid body rotation is illustrated in Figure 8: the U1 and U2 POM fields are anti-symmetric, present a very uniform distribution along the longitudinal axis and have equal participation factors to the POM. The third mode corresponds to barrel distortion even for the rotating speed of 1500 rad/sec. As the rotation speed is reduced, the singular value percentage is gradually accumulated to the first mode. Nevertheless, shaft rotation affects POD results even for the slow speed of 1.5 rad/sec, when these are compared to the corresponding results in Paragraph 3.1. On the other hand, the accelerating shaft response is totally different: only two modes allocate considerable amounts of energy. Both modes correspond to rigid body rotation. This shows that during the acceleration phase, the process is more consistent to the rigid body rotation than in the case of a rotating shaft.

These results show two important facts. First, the POD method identifies rigid body rotation, even when the rotation speed is quite slow. This is of importance in applications where the shaft – rather beam in such cases – should not rotate. On the other hand, this sensitivity to rigid body rotation degrades the POMs that correspond to deformation. In cases where the rigid body rotation is not required, removing it is done by appropriately transforming finite element results to a synchronously rotating coordinate system.

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