Multi-criteria analysis of elastic couplings with metallic flexible membranes.

Dobre, Daniel ; Simion, Ionel

1. INTRODUCTION

These couplings use clamping packs of spoked membrane to combine an adequate level of torque transmission with a reasonable axial and angular misalignment capability.

In order to transmit the torque between two shafts of a kinematical chain, in conditions of compensation of some important misalignments, it is necessary for coupling producers to handle some multi-criteria requirements regarding size, safety in exploitation (by guaranteeing a superior mechanical strength), dynamic behaviour and constructive simplicity.

The paper task is analysis of stress and buckling state of the metallic membrane as component of the flexible couplings. This task is carried out by a Finite Element Analysis (FEA).

2. ON STRESS STATE FOR MEMBRANE

The couplings having spoke form membranes transmit the torque from the inner diameter to the outer diameter and reciprocally, by shear loading of the membrane. The flexible membranes are made from spring steel and have thin spokes by making radial indents in the central portion (figure 1), the deformation of the spokes giving to the coupling its flexibility and thus its ability to handle installation misalignments. The couplings use these membranes arranged in a packet and assembled rigidly at their inner and outer diameters to other components. An example of this coupling is given in the fig. 2.

The mechanical strength analysis is made only for a single membrane, for the critical situation of existing axial deviations, taking into consideration that the stress as greater as in the case of deviation absence.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The stress state of the membrane spoke is determined by:

* torque transmission causing shear, bending and tensile stress;

* the angular and axial misalignments that causes bending and tensile stress;

* centrifugal loading.

For the membrane study by FEA the meshing is realised with finite elements of type SHELL, because the membrane depth is very small (0.3 mm), resulting 9054 nodes and 8316 elements. The meshing with the maximum size of 0.5 mm on the side resulted after a number of elements to ensure a sufficient convergence and accuracy of the FEA. The figure 1 shows the finite element mesh of the flexible membrane with trapezoidal holes using a uniform distribution of the elements.

The analysis was made for the nominal torque T=50 Nm and the rotation speed of 4500 rpm. A stress analysis was made for the cases without deviation and with axial deviation of 0.6 mm in the absence and with consideration of centrifugal load.

The tables 1 and 2 indicate the values for the principal stresses [[sigma].sub.1] and [[sigma].sub.3] and the maximum equivalent stress calculated using the von Mises theory [[sigma].sub.eq max].

One would see that the introduction of axial misalignment ([DELTA]a = 0.6 mm) increases almost three times the principal stress [[sigma].sub.1] and the maximal equivalent stress given the situation without axial deviation.

The equivalent (von Mises) stress distribution in the nodes of the structure is presented in figure 3 (with axial deviation of 0.6 mm) in grey colours. One would see that the maximum stress is located to the filleted root of the spokes. Other analysis at different torque (non-presented in this paper) shows that the maximal stress appears also on a large membrane area, that is the membrane solicitation is increasing with the load.

The maximum stress, [[sigma].sub.eq max], serves for the calculus of the safety factor, using the general expression:

c = [[sigma].sub.0.2]/[[sigma].sub.eq max], (1)

where [[sigma].sub.0.2] represent the yield point of the material. The Romanian steel symbol used in calculus is OLC 65A, having [[sigma].sub.0.2] = 800 N/[mm.sup.2]. The admissible safety factor indicated in literature for pieces subjected at fatigue (Niemann, 1986) is [c.sub.a] = 1.5 ... 3. The values of safety factor for the two cases of stress analysis and the discussion regarding the checking at stress solicitation are indicated in the table 3.

[FIGURE 3 OMITTED]

3. BUCKLING ANALYSIS BY FEA

Buckling analysis determines the loading level at which the membrane becomes unstable. This analysis takes into account the effects resulting from the compressive stresses, which tend to lessen the capacity of the membrane to resist lateral loads. As the compressive stresses increase, the resistance to lateral forces decreases, so that, at some load level, this negative stress stiffening overcomes the linear structural stiffness, causing the structure to buckle (Sorohan & Sandu 1997).

The linear buckling analysis determines the scaling factors, [c.sub.f], for the stress stiffness matrix, which offset the structural stiffness matrix. The equation that describes linear buckling is:

([K] - [c.sub.f] x [S]){u} = 0 (2)

where: [K] is the structural stiffness matrix; [S]-the stress stiffness matrix; [c.sub.f]-the buckling load factor; {u}-the eigenvector representing the buckled shape.

The buckling load factor is a safety factor, defined as:

[c.sub.f] = Limit buckling load/Applied load (3)

For the buckling analysis the membrane was blocked both on its inner and outer holes as displacement on the direction of the membrane axis with considering the centrifugal loading.

The buckling load factors for 10 shape modes has values in the interval [c.sub.f] [member of] [11.767, 17.625], being greater than the ones admissible conforming to Niemann (1986), [c.sub.a] = 3 ... 5. This means that the membrane is stable at buckling.

The figure 4 shows an image of the second buckling shape. The buckling load factor, [c.sub.f], is noted FREQ, a denomination used in the ANSYS 6.1 program.

[FIGURE 4 OMITTED]

4. MODAL ANALYSIS OF THE MEMBRANE

The modal analysis is necessary to extract the natural frequencies and mode shapes of a flexible membrane. A membrane should be designed to produce natural frequencies that will prevent the coupling component from vibrating at one of its fundamental modes under operating conditions.

The dynamic torsional stiffness determines the torsional vibration behaviour of the drive system. This gives rice to very large natural frequencies and the operating speeds lie below the resonant speeds. The natural frequencies for 10 vibration modes have values in the interval f [member of] [2013, 2577]. This means that the membrane is stable at vibrations.

It is relevant to state that the increased of dynamic torsional stiffness causes an increase in the resonant speeds (Birsan & Jascanu 1998). This is important because the resonant speeds should be present above the operating speed range.

5. CONCLUSION

The finite element analysis (FEA) is basic to evaluate stress and buckling state of flexible membrane.

The areas of maximum equivalent stress are placed on fillet root of the membrane spokes. The maximum equivalent stress values increase with the axial deviation at the limit of the recommended admissible safety factor values.

The centrifugal loading has a significant stiffness effect which causes law increasing of the buckling safety factors, leading to the increased resistance to torsional deformations and to deformations caused by the misalignments of the two shafts.

The results are important for other researches for the case of the entire membrane packet and are in concordance with the experimental dates presented by Dobre (2004).

6. REFERENCES

Birsan, J.G. & Jascanu, M. (1998). Elastic couplings' dynamic, Tehnica Publishing House, ISBN 973-31-1232-1, Bucharest

Dobre, D. (2004). Researches on multi-criteria optimization of elastic couplings with metallic flexible membranes, Ph.D. Thesis, University "Politehnica" from Bucharest

Niemann, G. (1986) Maschineelemente, Springer Verlag, Berlin

Sorohan, S. & Sandu, M.A. (1997). Nonlinear FEM-Analysis of a Diaphragm Spring, ELFIN 4, Iasi, oct. 9-11, pp. 125-128.

Tab. 1. The results of stress analysis for the case without
deviation ([DELTA]a=0), by considering the inertial loading.

Parameter

Principal stress [[sigma].sub.1] [N/[mm.sup.2]] 71.8
Principal stress [[sigma].sub.3] [N/[mm.sup.2]] -65.2
Maximum equivalent stress [[sigma].sub.eq max] [N/[mm.sup.2]] 69.7

Tab. 2. The results of stress analysis for the case with axial
deviation ([DELTA]a = 0.6 mm).

Parameter

Principal stress [[sigma].sub.1] [N/[mm.sup.2]] 286
Maximum equivalent stress [[sigma].sub.eq max] [N/[mm.sup.2]] 255

Tab. 3. The values of membrane stress safety factors.

 Analysis case

 Size Without With axial
 deviation deviation
 of 0.6 mm

Maximum equivalent 69.7 255
stress
[[sigma].sub.eq max]
[N/[mm.sup.2]]

Safety factor, c 11.48 3.14

Checking discussion c > [c.sub.a] c [congruent to] [c.sub.a]