Limits Of The Timoshenko Bar: FE Calculation Of Waves

2023 Author: Hannah Pearcy | [email protected]. Last modified: 2023-11-26 11:39

Ever shorter product development cycles require high-performance modeling approaches and calculation algorithms. The calculation approaches of the FVA Workbench are based on analytical methods that have been established in drive technology for decades and have been validated by FVA research projects. The advantage of these solutions is the high calculation performance with very good results

However, not all bodies can be described analytically with sufficient accuracy. Housing, planet carriers, wheel bodies and shafts in particular are therefore taken into account in the FVA Workbench using a finite element approach.

FE for complex geometries

The FE approach is generally suitable for complex component geometries that can no longer be mapped within the scope of the analytical approaches (Figure 1). This difference is explained below using the example of the shaft calculation newly integrated in version 5.6 of the FVA Workbench using the FE approach.

Calculation of the wave deformation

The wave deformation is calculated in the FVA Workbench using the Timoshenko approach. In this approach, the bending deformation according to the Euler / Bernoulli method is combined with the consideration of the shear deformation. The Timoshenko approach has the following limitations:

• The cross-sectional area of the component does not bulge.
• Only rotationally symmetrical components are calculated (solid and hollow shafts).
• Conical or curved contours are replaced by stepped cylinder sections.
• Forces and moments are introduced at the center axis.
• The power flow in stepped waves is not taken into account correctly.

With the majority of common shaft geometries, these restrictions do not lead to practical deviations from the real shaft deformation. However, if more complex geometries are used or if you want to check whether the limitations of the Timoshenko bar in a shaft geometry do lead to noticeable deviations, the FVA Workbench from version 5.6 can also calculate shaft deformations using FEM.

For this purpose, the waves set up in the FVA Workbench can be networked internally. With more complex geometries, waves exported from CAD programs can be read in and networked. The networking and determination of the contact nodes with the rest of the transmission model is largely automatic.

User meeting mechatronic drive technology

The focus of the user meeting mechatronic drive technology is on the mechanical components of gears, clutches and brakes as well as their design, dimensioning and interaction in the overall mechatronic system.

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To the FE wave quickly and efficiently

The user guidance has been designed for maximum efficiency so that an FE wave can be modeled, networked and calculated within a very short time. For internal networking, both linear and quadratic tetrahedral elements can be used for the network. With just a few clicks, every user can create a fully-fledged network for an FE calculation, even without special FE knowledge. This automated networking is possible because the deformation analysis carried out here places far less demands on the mesh fineness than a stress analysis.

User meeting mechatronic drive technology

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Calculation example: Comparison Timoshenko bar - FE method

The difference between the FE calculation and the Timoshenko calculation is explained below using the example of a stepped shaft. The comparison calculation was carried out on a simplified shaft geometry (Figure 2). The shaft has two bearings and is loaded with a single force in the middle.

• The outer diameter of the middle shaft section is changed for the comparative calculations.
• The ratios in the range between 1 (smooth wave) and 3.5 (very strong paragraph) are varied.
• The basic diameter of the shaft is 50 mm.

FE method in the FVA Workbench

In order to couple the rigidity of the FE elements with the analytical approaches, the rigidity is reduced to the coupling points. In the case of the shaft, the coupling points are bearings, gears, load application points or couplings. In the reduction, a stiffness matrix is determined for the coupling points, which describes the deformation behavior at the coupling points as well as the complete consideration of the entire FE component. Therefore only the crosspoints are visible in the calculation.

Calculation under the influence of all deformations

In a post-processing step, the deformations of the entire component can be calculated using the loads on the coupling points. This procedure enables a high-performance calculation under the influence of all deformations in the transmission. The influences on the gearing are experimentally demonstrated in the research project FVA 592 II.

Deviations of the two methods

Figure 3 shows the maximum shaft countersinking over the ratio of the outer diameter of the middle segment relative to the diameter of the adjacent shaft sections.

It can be seen that for a smooth wave the analytical approach gives the same result as the FE calculation. From a diameter increase of 1.25 times, the analytical solution calculates a lower countersink for this shaft geometry than the FE calculation. Starting with the 3-fold enlargement of the diameter of the middle section, the FE calculation shows a constant drop of 23% higher than with the analytical approach.

In this case, the difference can be attributed to two causes: on the one hand, it is due to the unevenly distributed force flow across the cross-section, and on the other hand, it is due to the curvature of the shaft cross-section in the area of the diameter step (Figure 4). As described above, these two effects are not taken into account in Timoshenko's analytical approach.

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Practical example

The described differences in the calculation method for shaft deflections can also be determined in practical gear models, such as the bevel helical gear shown in Figure 5.

Here, the face load distribution of the output stage was first carried out for the following variants for the intermediate shaft:

• Analytical calculation of all waves
• FE calculation of the intermediate wave, remaining waves were calculated analytically.

Figure 6 shows the load distribution for both calculated variants. Although here the influence on the shaft deflection is not as strong as in the theoretical example in Figure 2, the calculation of the intermediate shaft with FE shows a noticeable increase in the width factor KHß from 1.22 to 1.27.

Realistic representation of waves and outlook

Simultaneously with the implementation of the FE calculation of waves, the graphic representation of the waves in the 3D model has been significantly improved. As of version 5.6 of the FVA Workbench, the detailed geometry of notches such as feather keys, shaft shoulders with undercut and rectangular grooves is realistically displayed in the 3D model, so that the user receives graphic feedback on the geometry entries made (Figure 7).

These enhancements will also be carried out in the next version of the FVA Workbench with a view to the implementation of the FKM guideline for the calculation of shaft safety, which will then supplement the current shaft safety calculation in accordance with DIN 743.