MECHANICS. DIFFERENTIAL EQUATION SYSTEM OF ELASTICITY AND ITS BOUNDARY ELEMENTS PROBLEM
10. DYNAMIC ANALYSIS OF THREE-DIMENSIONAL BARS, DE- DE-TERMINATION OF NATURAL FREQUENCY USING PROGRAM DE-TERMINATION OF NATURAL FREQUENCY USING PROGRAM
10.4. The finite element solution of the task
2 1
0 c
(10.3)
where: - Θ the moment of inertia of the disks around Y axis - c0 the torsional spring constant,:
G I c a
p
0 (10.4)
where: -Ippolar moment of inertia -G - modulus of rigidity
On this basis, the torsional natural frequency is:
α0 =1503.87 1/s i.e. n=14360.9 rpm.
10.4. The finite element solution of the task
Structure shown in Figure 10.1 is a very simple geometric model, shaft can be characterized by a single line. We draw it as three separated line to help generating of the finite element mesh. So we can place the MASS elements on the end of the sections, on geometrical (key-) points. The drawn sections shown in Figure 10.2.
Figure 10.2. Creating a geometric model
After the creation of the geometric model, follows the describing the properties of finite ele-ment mesh. First we select the needed eleele-ment type (eleele-ment group) (see Figure 10.3), which is in our case the BEAM3D element.
Figure 10.3 Select the element type
In the next step we define the required material properties, the elastic modulus and the mod-ulus of rigidity (see Figure 10.4).
Figure 10.4 Define the material properties
Finally, the real constants are defined. The Figure 10.5 shows an example of simplified pro-cedures for definition the real constant by geometrical dimensions. The "2" sign indicate that the cross-section is circular.
Figure 10.5 Definition the real constant
If we have defined all properties of the finite elements, then we can create the finite element mesh (see Figure 10.6.). We create 10-10 element in each section. The section of BEAM3D elements is a circle, thus definition of the third node is not required.
Figure 10.6 Create the finite element mesh
We have to define the properties of the two disks. To this end, we define a new element group already described above, the MASS (inertial) element (see Figure 10.7).
Figure 10.7 Define the MASS element
To this element type does not belong to any material property, such as sufficient for definition the real constant. These constant of the first disc shown in Figure10.8. The moments of inertia of the disk around X and Z axis can be ignored, so their values shall be 0.
Figure 10.8 Real constant of first disk
The MASS element is placed on a single node in the finite element mesh. The creation of the MASS element shown in Figure 10.9.
Figure 10.9 Create a disc as finite element
We have to define the real constant of the second disc (see Figure 10.10).
Figure 10.10 Real constant of second disk
The creation of the second disc is similar to the previous one, just on another point of the geometric model.
The finite element mesh has five independent parts (the three shaft section and the two mass). We have to merge the common nodes to join these independent parts (see Figure 10.11).
Figure 10.11 Merge the common nodes
In the next step we determine the boundary conditions, shown in Figure 10.1 as bearings. This is similar to the previous examples, it can be defined fixing the three displacement degree of freedom at both ends of the shaft (see Figure 10.12.).
Figure 10.12 Define displacement constrains Thus the created finite element model shown in Figure 13.10.
Figure 10.13 The complete finite element model with the node numbering
Before the solving it is possible to set number of the calculated natural frequency (see Figure 14.10). It is appropriate to set calculate more harmonious, because we expect two-way bend-ing and torsional vibrations. In this study we will calculate the first 10 natural frequencies.
Figure 10.14 The natural frequency analysis settings After the setting follows the solution (see Figure 10.15)
Figure 10.15 Run frequency analysis
After a successful run the results can be displayed. The calculated first eight natural angular frequencies are shown in Figure 10.16.
Figure 10.16 The calculated natural angular frequencies
In the list, the first natural angular frequency is 10-5 1/s , which is negligibly small in the en-gineering practice. This is consistent with the learned in mechanics. The first natural frequen-cy of the multi-degree-of-freedom systems is zero. We observe that the 2-3. and 4-5. natural frequencies are the same. Later we will see that these two oscillation generated in X and Z directions. The 6. natural frequency has not pair. This is the torsional oscillation of the shaft between the two disks.
The finite element programs can display graphically the mode shapes as the deformed shape of the shaft (see Figure 17.10).
Figure 10.17 Display the mode shapes
The finite-element programs offer a scale factor to display the deformed shape. We override this scale factor and use 0,5 to do comparable mode shapes (see Figure 10.18)
Figure 10.18 The 2 4. and 6 mode shapes
In the figure, we observe that only one node belongs to the first mode shape. Also observed that in case 6 mode shape there is not visible deformation because the twisting around the Y axis is not visible in this representation
The displacements belong to 2. and 3. mode shapes are shown in Figure 10.19.
Figure 10.19 Values of 2 and 3 mode shape
The table contains very small magnitude displacements. These are not real values, only gener-ated during the solve as calculation errors.
The mode shape 6th is shown in Figure 10.20.
Figure 10.20. The 6. mode shape
The table contains only rotation results around the Y axis. It is also shown that the torsional oscillation can only be between the two disks.
10.5. Remarks
In engineering practice the torsional vibration analysis usually are used only a long, flexible shafts, flexible couplings.
The bending oscillation of rotating shaft with circle or pipe cross section may also be ex-amined using BEAM2D elements.
11. INTRODUCTION TO PLANE PROBLEMS SUBJECT. APPLICA-TION OF PLANE STRESS, PLANE STRAIN AND REVOLUAPPLICA-TION SYMMETRIC (AXISYMMETRIC) MODELS