Determination of Polar Moment of Inertia and Stress Concentration of Shafts Under Torsion Load with Arbitrary Cross Section

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Determination of polar moment of inertia and stress concentration of shafts under torsion load with arbitrary cross section

A. Fuerst, H. Sprysl

Hydro Generator Technology centre, ABB ALSTOM Power Switzerland

Abstract

In machinery shafts are important machine elements for power transmission. In structure and dynamic analysis for shafts or beams the stress concentration and critical speed have to be observed. To determinate the torsional vibration the polar moment of inertia of a cross section has to be known. This paper presents a method that is easy to use and very precise to get the polar moment of inertia and shear stress concentration of shafts with arbitrary cross sections. It presents the theoretical derivation of the thermal analogy and its application with Boundary Element Method (BEM). A comparison to analytical solutions shows the excellent accuracy of the method. Finally, the method is applied to a spider shaft of a big generator. Simulated and measured data is compared.

Introduction

The Boundary Element Method (BEM) together with thermal analogy can be used to determinate polar moment of inertia and shear stress. This shows significant advantage, in terms of precision and cost, compared to the direct application of the very popular Finite Element or Boundary Element Method. The handling of the method is superior to soap film or flow analogy. This paper presents the derivation and application of the thermal analogy employed with the BEM. A comparison with analytical solutions and measured data shows the excellent accuracy of the method.

110 Boundary Elements XXIJ Theoretical derivation of the method

The presented method to determinate the polar moment of inertia and stress concentration of shafts with arbitrary cross sections by thermal analogy is mainly thought for shaft with key ways, spline couplings, spider shafts etc. Under the consideration of a simply connected domain for the cross section the method of thermal analogy can be applied to determine two essential properties of shafts subjected to torsional load. At first, the equations for the shear stress and then the thermal analogy to solve in an efficient manner are derived.

The Laplace differential equation, eqn (1), describes the potential energy by the warp function 4* of an area subjected to the torque M%. This equation has to be solved under consideration of the boundary condition, eqn (2).

Figure 1: Cross section A of a shaft subjected to torsion

= 0, with x,ye A (1)

_^^ x,yeboundary (2)

To simplify solving eqn (1) one introduces the stress function O and the shear stress T, eqn (3) and (4), with the boundary condition in eqn (5). G describes the shear modulus of rigidity and 1} the angle of twist per unit length.

(3)

+ d O(x,y) ^ _2^ , %t the boundary <& = 0 (5)

The torque M% has to build a static equilibrium with the shear stress acting at area A, eqn (6). For a simply connected domain, like it is observed here, the resulting torque is quoted in eqn (7).

M, = J(T,yX - T,*y)dA (6) A

M, = 2jOdA (7)

Boundary Elements XXII 111 The polar moment of inertia of the cross section A is given in eqn (8).

Af, 2

A For structural analysis of beams or shafts subjected to torque the shear stress maximum, eqn (3) and (4) has to be known. For the analysis of the torsional vibration the polar moment, eqn (8), has to be determined.

To solve these problems the thermal analogy can be used. The advantage is that this formulation can be solved analytically for some simple cross sections. More complicated shapes can be solved with excellent accuracy at very low cost with BEM. By substituting the stress function O with the temperature T and 2G# with a quotient of source intensity q and conductivity X one reaches the thermal eqn (9) analogue to eqn (5).

2 = _.i, at the boundary T = 0 (9)

It is advantageous to choose the conductivity A, as 1 and source intensity q as 4.

The equation of the polar moment of inertia, eqn (8), turns out the integral of the temperature over the cross section, eqn (10). The resulting torque is twice the polar moment, eqn (11).

J, =JldA (10) A M, = 2J, (11)

The shear tension turns out the flux magnitude F, eqn (12).

9n 3n q 1^ 9n Concluding, to determine the shear stress and polar moment of inertia of a cross section one applies a conductivity X of 1 and source intensity q of 4 and read out the flux magnitude F and the integral of the temperature.

Comparison of analytical solution with the application of thermal analogy with BEM

Application to a square cross section

The thermal analogy has been used for a couple of examples. The first one is a square cross section with a = 100 mm, see figure 2. The calculated polar moment of inertia is 1.4058 E7 mn/ and the highest shear stress is 135.07 N/mnf. This result is similar to commonly used solutions like in Dubbel [1] Jt= 0.141a^= 1.41 E7 mm* and T = MJ W,= 2 * 1.4058E7 / 0.208 * 100*= 135.173 N/nW.

112 Boundary Elements XXII

The performance of the thermal analogy is clearer if one compares it with the analytic results. Eqn (13) is the Poisson equation. Its solution, eqn (14), defines the polar moment of inertia.

AO = -2GO (13) 1 (2i + l)Jlb -tanh = 1.40577 E7,a = b = 100 (14) ' 3 TC* ^(2i + l)' 2a To complete the analytical observation the solution eqn (8) is employed to

determine of the shear stress T, eqn (15), and the torsional section modulus Wt, eqn (16).

1 1. 33720e+002 |l.2l712e*002 |l.09704e+002 9.7G955e+001 8.5G874e+001

; *6.16712e*001 1 4.96631 e+001 |3.76550c+001 #2.5G4G9e+001 ll.3638Be+001 Il.63069et000

Resultant ilux magnitude Max= 135.07

Figure 2: Thermal analogy solution for a square cross section with a = 100 mm, using double symmetry

1 (15) l)7ib

= —L = 208165,witha = b = 100 (16)

Therefore T^ax = 2 Jt / Wt equals to 135.063. The concordance between the

analytical results, obtained with Mathematica, and the results by thermal analogy, obtained by BEASY, are excellent. They match well with the results in literature too.

Boundary Elements XXII 113

Application to a round shaft with a circular key way

The next example is a circular shaft with a radius R = 100 mm, a circular key way of radius r = 10 mm and centre point at the boundary, see figure 3. The calculated results by BEM are 379.99 N/nW for the maximum shear stress and Jt - 1.54166 E8 mm* for the area polar moment of inertia that match very well with the analytical solution. The solution of the Poisson differential equation describes the stress function O, eqn(17). The cross section is rotated 90 degree for this formulation.

(17) 0(x,

One can deduce the formula for the maximal shear stress and the polar moment of inertia.

% = GO(2R - r) = 380N / rrmf (18)

J.=- --Ttr I—**I|=1.5419E8mm« (19) 4 3it

13.761906*002 3.42372e*002

'3.08554e*002 2.74735e*002 2.

13.80061e*001 |4.18769e*000 Resultant flux magnitude Max= 379.99 Min= 0.39174

Figure 3: Circular shaft with circular key way

114 Boundary Elements XXII

Application to a round shaft with a rectangular key way

The next example is a circular shaft with a common keyway, see figure 4. The radius of the shaft is R = 5.95 mm, the width of the keyway 4 mm and the depth 2.4 mm. The radius in the keyway is 0.14 mm. The maximal shear stress

calculated by BEASY is T^X = 29.202 N/mnf in radius r = 0.14 mm. The area polar moment of inertia is J^= 1438.5 mm\ The nominal shear stress is given in eqn (20). 0 T 1 (\ (20)

The comparison of the average with the maximum shear stress shows for the

stress concentration factor 04 = 3.36. In the literature one finds for the same example a FDM solution with 04= 3.4, see H. Parkus [2]. But this value varies, see f.e. Cornelius and Jager [3] or Dietmann and Ross [4].

2.89104ct001 2.63130e+001 2.371566+001 '2.111832+001

1.85209e+001 % 1.592356+001 h.33261e+001 |1.072886+001 |8.13139e+000 |5.53402e+000 |2.93664e+000 §3.392656001

Resultant flux magnitude Max= 29.202 Min= 4.77232E-02 Figure 4: Round shaft with rectangular keyway

Application to a spider shaft of hydro generators

The present procedure can be very useful for calculation of vibration problems with non-circular shaft cross sections, for example within turbo or hydro generator power plants, see figure 5 and figure 6. The natural frequency of torsional vibration for the shaft according to figure 6 and with the spider cross

section according to figure 5 was calculated and measured. The value was close to 50 Hz. For electrical machines this frequency is very dangerous f. e. in the

Boundwv Elements XXII 115 case of terminal short circuit. The shaft stiffness could not be increased and the

shaft was changed with a cross section according to figure 6. The spider shaft in figure 5 has a kernel diameter of 620 mm and outer diameter of 1040 mm. The shaft in figure 6 has a kernel of 625 mm and an unchanged outer diameter. In our opinion it is very difficult to estimate the equivalent diameter for the shaft

without such BEM calculation. The area polar moment of inertia calculated by BEASY Jt= 2.47133E10 mrn* can be recalculated to equivalent diameter D^ = 708.31 mm for figure 5. This value has been used in the vibrations calculation. The new replaced shaft was measured once more. The natural frequency of torsional vibrations was shifted 10 % below possible excitation.

The very small potential error norm of maximum 0.024 at the inner radii for this BEASY calculation shows the quality of the solution at the boundary. For good results of polar moment of inertia that is analogue to the integral of potential, eqn (10), it is necessary to set sufficient internal points into the cross

section.

I1.24833C+003 , ,1.13G02e+003 jfl.02370e+003

,:. 7.99078C+002 j||B.86765e+002 %5.74453et002 • 4.62140e+002

13.496276+002 12.37515e+002 |l.25202e*002 |l.28894e+001 Resultant (lux magnitude Max= 1260.9

Figure 5: Cross section of a spider shaft

Figure 6: Spider shaft of a hydro generator

116 Boundary Elements XXII

Conclusion

This paper presented the thermal analogy applied with BEM to determine polar moment of inertia and the shear stress of simply connected cross sections. The method shows excellent accordance to results in literature and analytical solutions. Finally it is employed for a dynamic study of a spider shaft of a hydro generator.

References

[1] Dubbel Handbook of mechanical Engineering, New York, p. B29, 1994 [2] Parkus, Osterr. Ing. Archiv 3., 336, 1949 [3] Cornelius and Jager, Konstruktion 22, 5, 1970

[4] Dietmann and Ross, Konstruktion 22, 9, 1970.