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# Mechanics

The "Projects and Calculations" labs (BPC) has two main roles : achievement of innovative projects and carrying out complex calculations in many physical fields for all ILL divisions.

# Mechanics

The BPC can perform very complex mechanical calculations on samples and obtain detailed information on stress, displacement, eigenfrequencies, etc.

# Structural calculations

This example presents a deformation calculation for the IN16B movable aluminum vessel (under vacuum). Its internal volume is roughly 35m3 and its weight 8 tons

# Disk geometry optimisation for the NIST institute

### Thickness profile

The optimal profile for a disk with no windows is a Gaussian profile. However the problem becomes more complicated for disks with windows.

The windows destroy the axisymetric equilibrium between the radial and hoop stresses and essentially "cut" the continuity of the hoop stresses. This leads to a concentration of stresses at the lower part of the windows.

To optimise the design we have to find a way to minimise the stresses in this region.

To solve this problem we have been investigating another profile, which would allow the mass of material to be reduced at the extremity of the disk, after the fillet.

The optimal solution is a hyperbolic profile.

The thickness of the central area of the disk is kept constant in order to optimise its mass; the total thickness is maintained at the centre, given that the stress is at its maximum in the central hole.

### Stress and deformation analysis

Method :

This disk was modelled using the finite element software Ansys.

Due to its cyclic symmetric structure and symmetry plane, only a portion of the disk (1/24th) is represented.

The model consists of SOLID 45 3D elements, and SHELL43 2D elements which model the gadolinium.

The mesh is refined at the areas subject to maximum stress (the window’s fillet and central hole).

The disk is submitted to a rotational speed of 18 000 revolutions per minute.

Results of the stress study:

For a periphery thickness of 2 mm, the maximum equivalent stress (Von Mises stress) is 258 Mpa in the fillet of the window. The security factor compared with the yield strength is about 2.

The study of the principal stresses shows that circumferential stresses sq are predominant compared with radial stresses (259 Mpa versus 158 Mpa).

# Example of a modal analysis of choppers

We performed a modal analysis to monitor the potential vibrations of the disks. The excitations known are the residual imbalance and the shaft's longitudinal active magnetic bearing.

Experience on similar choppers has shown that any excitation is predominantly due to the longitudinal active magnetic bearing. The modal analysis therefore focussed on the eigenmodes along the z axis.

Method

The mesh of the model is specific: mapped mesh, 20-node tetrahedral elements SOLID186.

However the disk presents cyclic symmetry, and the modal analysis must be performed on the entire disk, although only a small part of the spectrum to be searched is to be calculated.

We will study the eigenfrequencies whose modal mass is significant compared with the total mass of the disk.

The boundary conditions are:

All degrees of freedom are fixed for 0 £ r £ 28.5 mm.

### Study of the eigenfrequencies at a standstill

The first eigenfrequency of the disk occurs at 395 Hz. It is a rotation mode about a w axis (in the xy plane). However the effective mass corresponding to this mode is very small: only 0.1% of the mass of the disk.

Eigenmodes along z:

• The first eigenmode with a significant modal mass is at 451 Hz. It is a corolla mode. Its effective modal mass is about 2.56 kg, which represents 41.6% of the total mass of the disk.

• The next significant eigenmode is at 942 Hz. Its effective mass is about 2.2kg (36% of the total mass).

The torsion mode around the z-axis is at 565 Hz.

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