6 Model-Updating in Non-Rotating Condition Using Modal Analysis
6.1 Single Components
6.1 Single Components
In what follows, the steps to verify, validate and update the model representation of the diamond core bit will be described as an example of the rotating components modeled as elastic bodies.
On the one hand, the diamond core bit appears to be a rather simple structure in the sense of structural dynamics, consisting mainly of a hollow tube. On the other hand, the geometry at the connecting end towards the chuck is rather complex, and the whole component is made out of four different materials. The very tip is made out of a sinter metal, containing the industrial diamonds as cutting material, the actual tube is made out of steel, and the connecting end consist of aluminum which is mounted around the steel tube in a casting process. For a 3D solid modeling approach, none of the before mentioned things is imposing a real challenge. The core bit is decomposed into different sections and the corresponding material parameters are applied. At the boundaries between the different sections, tie con-straints are introduced, while at the same time the mesh is setup such, that the nodes are congruent within the joint patches as far as possible.
6.1.1 Mesh-Grid Convergence Study
To verify the numerical solution, a mesh-grid convergence study is performed using the natural frequencies of the first ten modes in free boundary conditions as a criterion to evaluate the error. The study starts with a very coarse mesh, which is refined in subsequent steps until the numerical solution converges. Important parameters for the mesh refinement are the element size and the number of elements per corner arc. Since the lower modes are completely dominated by the tube-like part of the core bit and since the connecting end needs a much finer mesh anyway due to the contact formulations with the chuck, the mesh is changed only on the tube. Table 6.1 contains an overview of the results, while Figure 6.1 shows different levels of mesh refinement as well as pictures of typical mode shapes.
The first ten modes contain four different types of modes: Bending modes, ovalization modes, which are typical for a hollow structure, and finally a torsional and even a longitudinal mode.
Both the bending and the ovalization modes appear as double modes due to the (cyclic) symmetry of the structure. The largest influence of refining the mesh shows itself in the ovalization modes, while other mode types are less affected. After the fifth iteration, there are no more recognizable changes in the natural frequencies and the solution has converged.
6 Model-Updating in Non-Rotating Condition Using Modal Analysis
Table6.1:Model-updatingofthecorebit modeshapeandnaturalfrequencies 1st1st2nd2nd3rd1st4th5th3rd1st bendingovalizationovalizationbendingovalizationtorsionovalizationovalizationbendinglongitudinal measurement naturalfrequency[Hz]40896010541115126012931454176720652145 dampingξi%0.080.10.080.080.05-0.050.060.07- simulations 1elementsizen.elementsp.cornerarc:2[Hz]414102611201128132313001509184220862157 rel.error%1.46.96.31.15.00.53.84.31.00.6 2elementsizen/2.elementsp.cornerarc:3[Hz]414100610991128129813011482181120872158 rel.error%1.44.84.31.23.00.62.02.51.10.6 changestopriorsolution%0.1-2.0-1.90.1-1.90.1-1.8-1.70.10.0 3elementsizen/4.elementsp.cornerarc:4[Hz]414100210951128129213011475180220872158 rel.errorin%1.44.43.91.22.50.61.52.01.10.6 changestopriorsolution%0.0-0.4-0.40.0-0.50.0-0.5-0.50.00.0 4elementsizen/8.elementsp.cornerarc:8[Hz]41499910911128128813011470179520862157 rel.errorin%1.44.13.51.22.20.61.21.61.00.6 changestopriorsolution%0.0-0.3-0.30.0-0.30.0-0.3-0.40.00.0 5elementsizen/16.elementsp.cornerarc:12[Hz]41499910911128128713011470179520862157 rel.error%1.44.13.51.22.20.61.11.61.00.6 changestopriorsolution%0.00.00.00.00.00.00.00.00.00.0 6sameas#3.updatedouterdiameter[Hz]41396610611127126112951444177820862149 rel.error%1.20.60.71.10.10.2-0.70.71.00.2 7sameas#6.E-modulereducedby1.4%[Hz]41095910541119125212861434176620712133 rel.error%0.5-0.10.00.4-0.6-0.6-1.30.00.3-0.5
6.1 Single Components
(a) simulation #1 (b) simulation #5
(c) 1st bending mode (d) 1st ovalization mode
(e) 1st torsional mode (f) 1st longitudinal mode
Figure 6.1: Varying mesh grid and mode types of the core bit
6 Model-Updating in Non-Rotating Condition Using Modal Analysis
In the further course of this thesis, model #3 is used to simulate the run-up of the rotor system.
The next section will discuss validating and updating the simulation model.
6.1.2 Model-Updating Using EMA
Table 6.1 also contains the results of a validation experiment in the form of EMA, performed according to Chapter 4.2.2. A comparison to iteration #5 of the simulations reveals that although the numerical solution has converged, there are still deviations of more than four percent between simulation and experiment. The model quality might profit from analyzing and possibly eliminating the causes of variance.
Taking a closer look at the results of the mesh-grid convergence above reveals that the natural frequencies of all ten modes are higher in the simulation than in the experiment, regardless of mesh refinement. This is an indication that the cause of the deviation most likely lies in global parameters, such as geometric or material properties. At the same time, the magnitude of deviation is not the same over the four different mode types. In ovalization modes, the deviation is significantly higher than with the other mode types. Taken together, the two findings imply that the discrepancies between simulation and experiment are most likely caused by deviations in more than one global parameter. A straightforward way to validate an FE-model intended for dynamic analysis, is to check the mass. In this case, the comparison between the model and the physical component of the core bit revealed, that the model shows a slightly higher mass. The next step consists of checking the geometric dimensions of the physical core bit. The outer diameter of the steel tube turns out to be a few hundreds of a millimeter smaller than the nominal value used to build up the model, still being within the specified manufacturing tolerance, though. While the deviation is minimal, in such a slender cross-section as that of the steel tube, it has a noticeable influence on the natural frequencies. Simulation #6 contains the updated geometry and already shows a good improvement when compared to #3 and even in comparison to #5. A method to include uncertainty in geometric parameters in the calculation of natural frequencies is presented in [123] using a bladed rotor as an application example.
The correlation between experiment and simulation can be further improved by adapting the Young’s modulus of the steel tube. In simulation #7, the Young’s modulus is changed by approximately 1.4 % from 210 GPa to 207 GPa, resulting in an almost perfect match compared to the experimental data. According to Langer et al. [78, 79], 1.4 % deviation in
6.1 Single Components
the Young’s modulus is a reasonable range of uncertainty.
Another important parameter to validate is the damping. The current simulation model considers damping in the form of material damping and discrete viscous dampers, as in the model approach of the bearings (see Chapter 5.2.2.1), or to account for the influence of the human operator (see Chapter 5.2.6). Besides the natural frequencies, Table 6.1 also contains experimentally identified modal damping ratios ξi, expressed as a fraction of crit-ical damping. In the following discussion, one should keep in mind, however that impact testing is not known to produce very accurate results with regard to modal damping. The nominal difference between the values, ranging from 0.05 % to 0.1 %, should therefore not be overestimated. A careful assessment comes to the conclusion, that the damping does not differ much between the different modes, and is in good agreement with generally accepted values [49, 152], stating a material damping of 0.1 % for steel as well as for aluminum.
ABAQUS offers several possibilities to consider material damping. In the current case, it makes most sense to assume Rayleigh damping, which treats the overall damping as a combination of mass proportional (low frequency range) and stiffness proportional (high frequency range) damping. This approach requires one to specify two Rayleigh damping factors, αR for mass proportional damping and βR for stiffness proportional damping. The modal damping ratios and the two damping factors stand in the following relationship [122]:
ξi = 1 2
αR
ωi +βRωi
(6.1) in which ωi stands for the natural (circular) frequencies. Once the modal damping ratios ξi are known from EMA, the two damping factors αR and βR can be identified by solving the overdetermined system of equations by using a least-square-error method.
As an interim summary, what can be noted is that modal analysis is a powerful tool to verify and validate the model representation of single components. Analyzing the cause of possible deviations and updating model parameters where necessary can significantly increase the confidence in the model. Verifying and validating the model from the bottom up, significantly reduces the sources of possible deviations when now turning to the full model.
6 Model-Updating in Non-Rotating Condition Using Modal Analysis