TOB-NOTE 00. 4Big Wheel Modal Analysis Using FEM Harri Katajisto July 25, 2000

1 Introduction

There were two reasons why the FE modal analysis of the BW was not possible toperform with the model used in the static analysis. Firstly, the number of calculationoperations is considerably higher in the modal analysis compared to the static analysis.Secondly, symmetry conditions are not recommended to use in the modal analysis ofmechanical structures.

An approximate model was built where the disks of the BW were modelled as solidcomposite circular plates. Before that the modal analysis was performed for the precisedisk model. By using the analysis result (lowest natural frequency) and well-establishedformulas for modal analysis it was determined the mass and stiffness for the approximateplate.

2 FE analysis of the precise model of the disk

Modal analysis was possible to perform for the precise disk model. For the analysis thedisk was clamped from its complete inner circumference (Fig.1 ). Mechanical propertiesof the ply were tailored so that the in-plane moduli of the laminate (E_x and E_y) were47 GPa, which corresponded to the material tests.

Figure 1. Boundary conditions used in the disk modal analysis.

The lowest natural frequency of this structure was 17,643 Hz, and the total mass was15,295 kg. The first mode shape is presented in Figure 2.

a

b

Figure 2. The first mode shape (symmetric) of the disk when clamped from the inner radius.

3 Approximate solution for the disk

In the approximate solution the BW disk is replaced by one solid round plate with an holein the centre. The outer and inner radius were kept the same as in the BW disk. Thematerial for the approximation has to also be orthotropic, because the bending stiffness ofthe skin radically depends on the orientation angle. This character (E^f_x`) is illustratedusing ESAComp software in Figure 3. In the same figure it can also be seen that the in-plane modulus remains constant, as the laminate is quasi-isotropic kind.

Figure 3. Laminate in-plane and bending stiffnesses as a function of orientation.

Several disk modal analyses indicated that the skins of the disk behave like two separatestructures, which are enforced to have same displacements. If the disk would act like asandwich structure the bending stiffness would not be an orientation dependent.

Next the density for the solid plate was defined. The outer radius of the disk is 1,2 m andthe inner radius 0,69 m Thus, the area of the plate is A = �(Rout2-Rin2) = 3,028 m2. Inthis approximation the used thickness (h) was the skin thickness (2,0566*10-3 m). Hence,the density of the solid plate was � = m / V = 2455,94 kg/m3.

The stiffness (Young’s modulus) for the solid plate was calculated basing on equation (1)

where

�mn = 2 � f�mn, a table constant that follows from the geometry and boundary conditionsa, outer radius of the circular plate (Fig.1)

m = � h [1, 2]

It should be noted that this equation is valid for isotropic cases. However, in this case itwas applied to the orthotropic plate. From this equation one can conclude that if thenatural frequency is divided by two, the stiffness must be divided by four. Same plymaterial (E1 = 122,5 GPa, E2 = 8,2 GPa, G12 = 4,2 GPa, and � = 0,3) was selected for thebase material and the analysis yielded for the lowest natural frequency 3,9141 Hz.Young’s moduli have to be increased so that the target frequency of 17,643 Hz isachieved. We got the following equations (2.a-c):

Thus, the Young’s modulus for the solid plate is given by (3)

)1(12 2

3

2

2

vm

Eh

amn

mn�

�

��

643,172

9141,3�

x

643,17

9141,3log2log �

x

1723,22log643,17

9141,3log

���x

GPaE

E 96,24884 1723,2

01

1 ���

Respectively, the other engineering constants were

When using these engineering constants and density � = 2455,94 kg/m3, it was verifiedthat the first two lowest natural frequencies were both exactly 17,643 Hz. The symmetricshape mode is presented in Figure 4.

Figure 4. The first shape mode (symmetric) of the disk when clamped from the inner radius.

4 Extension to the full-dimensional model with simplifications

Now the essential parameters for the BW modal analysis were defined. Panels and theinner cylinder were modelled as in the static analysis. The BW was supported from theinner cylinder. In a single point in the inner cylinder (where the cylinder and the disk areattached to each other; y = Rin) the global z-directional displacement was also prevented.The element type used in the calculations was the Linear Layered Structural Shellelement. This is a Mindlin-type element degenerated from a solid element. Same elementtype was also used in the static analysis. Three lowest natural frequencies for the BWwere: 23,8 Hz, 24,8 Hz, and 31,5 Hz.

The first three shape modes are presented in Figures 5-7.

GPaE 61,1662 �

GPaG 34,8512 �

Figure 5. The first shape mode of the BW (23,8 Hz). In this shape mode the inner cylinder is immobilized,the panels move rigidly whereas the disks vibrate like drum skin.

Figure 6. The second shape mode of the BW (24,8 Hz). In this shape mode the inner cylinder isimmobilized, the panels move rigidly whereas the disks oscillate around the disks vertical axis.

Figure 7. The third shape mode of the BW (31,5 Hz). In this shape mode the inner cylinder is immobilized,the panels move rigidly whereas the disks oscillate around the disks horizontal axis.

5 Conclusions

The quantity and distribution of the stiffness and mass are the major factors thatcontribute to the dynamic behaviour of a structure. The quantity and distribution of themass of the disk remained in the approximate solution.

The starting point for the stiffness evaluation was that the disk is clamped from itscomplete inner circumference, which seemed to be the case also in the BW structure. Themode shapes of the disk and the approximate disk were verified to be alike.

The third factor in modal analyses is boundary conditions. The BW is supported so thatrotations are free. This kind of stress free support decreases the natural frequencies of asystem. The reliability of the analysis results still needs experimental verification.

6 References

1 Pennala E., Koneiden ja Rakenteiden Värähtelyt, Otatieto, Espoo, 19972 Blevins D., Formulas for Natural Frequency and Mode Shapes