Critical Speed Analysis of Fibre Reinforced Composite Rotor Embedded with Shape Memory Alloy Wires

In the present analysis, the fundamental natural frequency of a Jeffcott and a two-mass rotor with fibre reinforced composite shaft embedded with shape memory alloy (SMA) wires is evaluated by Rayleigh's procedure. The flexibility of rotor supports is taken into account. The effect of three factors, either singly or in combination with each other, on rotor critical speed is studied. The three factors are: (i) increase in Young's modulus of SMA (NITINOL) wires when activated, (ii) tension in wires because of phase recovery stresses, and (iii) variation of support stiffness by three times because of activation of SMA in rotor supports. It is shown by numerical examples that substantial variation in rotor critical speeds can be achieved by a combination of these factors which can be effectively used to avoid resonance during rotor coast up/down.


INTRODUCTION
Shape memory alloys (SMA) have been commer- cially available since the past few decades, but their applications have been very limited, i.e. force and displacement actuators.Rogers (1990), suggested that SMA fibres could be embedded into conven- tional composites such as graphite/epoxy to control the structural response including static deforma- tion, vibration, buckling and structural acoustic radiations and transmission.The behaviour of SMA is governed by the diffusionless phase transformation between a high-temperature low- strain austenite phase and a low-temperature high- strain martensite phase.The shape memory effect arises from the interplay of temperature and stress in the free energy of the alloy.A common SMA is 55-NITINOL, which is a nickel and titanium alloy.Its Young's modulus increases about four times when heated above its austenite transformation temperature.Also it can be trained to have a particular shape while in austenite phase.If it is Tel." 0091-11-6861977.Ext.3166.Fax: 0091-11-6362037.E-mail: kgupta@mech.iitd.ernet.in.

202
K. GUPTA cooled to its martensite phase and subjected to plastic deformation, it will return to its 'memorised' shape when heated above the austenite transforma- tion temperature.
Recent developments (Singh et al., 1997) in Composite rotors have opened up tremendous possibilities of developing smart rotors by embed- ding SMA wires in fibre reinforced composite shafts.
Baz and Chen (1993) have analysed a composite rotor embedded with NITINOL wires by theoretical and experimental means.Their results have shown a reduction of about 50% in vibration amplitudes by activating NITINOL wires.Nagaya et al. (1987) proposed the use ofSMA in rotor supports to control the rotor critical speeds.
In the present work, a rotor made of fibre reinforced composite shaft embedded with NITINOL wires is analysed theoretically to estimate their effect on the rotor critical speed.A flexibly supported simple rotor with two identical masses mounted symmetrically about the midspan is considered.The effect of increase in stiffness (Young's modulus) and tension in wires due to phase recovery stresses when wires are heated to austenite transformation temperature, on the rotor critical speed is studied.In addition to these, the effect of variation of support stiffness due to activation of SMA in rotor supports is also studied.The objective of such a study is to estimate the amount of shape memory alloy in wires as well as in rotor supports, required to alter the rotor critical speed sufficiently enough (say by 30%) in order to avoid resonance during rotor coast up/down.Results have been presented in terms of non-dimensional parameters and illustrated through numerical examples.Some issues associated with activation strategies to be adopted for such rotors during coast up/down have been discussed in a qualitative sense.

THEORETICAL ANALYSIS
Consider a flexibly supported composite rotor (Fig. 1) embedded with shape memory wires and carrying discs in the form of lumped T strain and kinetic energies, U and V, respectively, are given by U -Eclcw,2xx dx + - For estimation of natural frequency, the shaft deflection can be assumed to be harmonic, i.e., w(x, t) W(x)cos pt.On substituting in Eqs.
(1) and (2) and following Rayleigh's method (Rao and  Gupta, 1984), the time dependence cancels out and the frequency equation is obtained.
A two-mass rotor with flexible isotropic supports as shown in Fig. is considered.The two masses are identical and mounted symmetrical about the midspan.From the static deflection of the system, the following mode shape is assumed: (1 + rt/rc2) In Eq. (3), m is mass of the shaft and El= Eclc + Enln.From Rayleigh's method, the following CRITICAL SPEED ANALYSIS 203 frequency equation in terms of the non-dimensional parameters is obtained" where It can be seen that natural frequencies for various special cases can be obtained from expression (4).If  ( is set to zero, we get the expression for a Jeffcott rotor with flexible supports and lumped mass 2M at the midspan.In addition, if rk 0, the expression for Jeffcott rotor on simple supports is obtained.If rm and r, are also set zero, then the RHS of expression (4) reduces to unity.
To estimate the effective longitudinal modulus, consider the cross-section of the shaft (Fig. 2).Here dh and dw are the diameters of the rubber sleeve and the SMA wires respectively.The stiffness of the rubber sleeve is ignored in the present analysis.The volume fraction of wires Vn and that of rubber Vr, the shaft mass per unit length m and equivalent longitudinal modulus E are given by nr2dw nr2dw (rhn 1) (6) rn--r-1' rr--r-1 m 1-t-(P-c--1)Vn-+-(Pcc-1)Vr, (7,   mc   FIGURE 2 Rotor cross-section.
The expressions for rnc and the various non- dimensional parameters in Eqs. ( 6 The longitudinal equivalent modulus of the fibre reinforced composite shaft is given by Tsai and   Hahn (1980) Here is the total thickness of the laminate equal to (do-di)/2 in the present case and ti and Oi are the thickness and the fibre angle of the ith lamina, N being the total number of laminae.The laminate variants U1 to U5 are given by U1 (3Qxx + 3Q + 2Qx + 4Qss), u3,5 1 / 2 (Qxx + Qyy + 2Qx 7: 4Qss), u4 1 / 2 (Qxx + Qy Qxy + 4Qss), (3) where the fibre and the matrix.

Gm
Es (Vm + VfGm/Gf)' /S =(Vm + x /m gm --/f gf. (5) |n the above, subscripts m and f denote the matrix and the fibre, respectively.The fundamental natural frequency of the composite rotor embedded with SMA wires can be determined by successive use of expressions ( 15) in ( 4).It may be noted that since the gyroscopic effects are not taken into account and support stiffness K is assumed to be independent of rotor speed, the natural frequency from expression (4) will give the first rotor critical speed.
Qxx (1 #2xEy/Ex QYY (1 #xEy/Ex) The longitudinal modulii Ex, Ey, the shear mod- ulus Es and the longitudinal Poisson's ratio #x, with respect to the principal material axes of symmetry, x and y (Fig. 3), are functions of the individual properties and the volume fractions of Y 2 X NUMERICAL EXAMPLES Figure 4 gives the equivalent longitudinal modulus of composite shaft as obtained from Eqs. ( 11)-( 15) for three types of composites boron/epoxy, carbon/ epoxy and glass/epoxy.The curves with solid lines represent single ply configuration with fibre angle denoted on x-axis.The curves with dotted lines denote a symmetric configuration (0, 45 , 45 , 0) with four layers (N= 4) of equal thickness and 0 varying from 0 to 45.Following typical material properties from literature (Jones, 1975)   assumed: The value of Young's modulus E for glass, carbon, boron and epoxy is taken, respec- tively, as 72.4,231,393 and 3 GPa.Poisson's ratio for glass, carbon and boron is assumed to be 0.2 and for epoxy as 0.3.The shear modulus is given by G=(E/2)(I+#).Assuming fibre and matrix volume fractions of 0.6 and 0.4, we get from expression (15) for glass/epoxy, carbon/epoxy and boron/epoxy, respectively, the value of Ex as 44.64,   140 and 237 GPa, the value of Ey as 7.06, 7.35 and 7.42GPa, and value of Es as 2.73, 2.83 and 2.85 GPa.The Poisson's ratio #x for all the three composites works out to 0.24.
Figure 5 is a graphical representation of Eq. ( 8), which is a linear relation between E/Ec and re--En/ Ec.In Fig. 5, plots are given for two values of rhn and 4 and four values of NITINOL volume fraction Vn=2%, 5%, 10% and 20%.The E/Ec ratio increases with increasing value of re and Vn.The value of E/Ec is also quite sensitive to the nondimensional ratio rhn.
From Fig. 4 it is noticed that the value of Ec could vary in a wide range from about 10 to 237 GPa.The value of En could vary typically (Rogers, 1990), from about 25 GPa in the unactivated martensitic phase (below the martensitic transformation temperature) to about 90GPa in the activated austenite phase (above the austenite transforma- tion temperature).Thus the range of re is from 25/237 =0.1 to 90/10=9.For fibre angles greater than 45 , value of Ec could be much smaller re- sulting in an upper value of re much greater than 9.
The variation of natural frequency (Eq.( 4)) with various nondimensional parameters rt, rk and rm is given in Fig. 6(a) and (b) for two values of 0 and 0.5, respectively.The effect of rm and rk is obvious.
The natural frequency decreases as the mass ratio rm is increased.Same trend is observed when r Ks/K is increased, i.e., the supports are made flexible.
In order to assess the effect of NITINOL wires on the natural frequency, it is necessary to consider two effects: (i) variation in the value of En from about 25 GPa in unactivated state to about 90 GPa in activated state as the temperature is increased from below the martensitic transformation temperature to one above the austenite transforma- tion temperature; (ii) development of large phase recovery forces in the wires when these are activated.The first effect is taken into account through Fig. 5, from which the equivalent longi- tudinal modulus E of the shaft is obtained.The second effect manifests itself through the non- dimensional parameter r, in the term (1-r/vr2) in the frequency equation (4).If the wires are imparted an initial strain and are suitably restrained, then large phase recovery tensile force T will develop due to shape memory effect when the wires are activated by increasing their temperature above the austenite transformation temperature.The phase recovery stresses (Rogers, 1990) can be as high as about 500 MPa.It should be noted that the value of r, could vary over a large range depending on relative values ofE, 1, do, di, Vn and T. Also for small values of rk (flexible shaft and rigid support), the effect of r, is more dominant because the shaft being flexible undergoes appreciable flexure.On the other hand, for large values of r (rigid shaft and flexible support), the flexure in the shaft is negligible and the effect of r, is small (see Eq. ( 1)).
fibre angle equal to 30 and fibre volume fraction of 0.6.The number of wires be N 18.Let the value of En be 25 and 90 GPa, respectively, in unactivated martensitic and activated austenite phases.
From Fig. 4, Ec 37 GPa.The volume fraction (Eq.( 6)) for wires is Vn 0.02 (2%).The values of re for the unactivated and the activated states of the wires are 0.676 and 2.43, respectively.The non- dimensinal ratio rhn 4. From Fig. 5 the values of E/Ec for these two cases is 0.9335 and 0.969.It is obvious that due to small volume fraction Vn of wires, the effect of these on equivalent stiffness E is negligible.Also because rhn> rE, there is a net reduction in the value of equivalent stiffness.This reduction will be more if Vn is increased.For example if Vn 20% the values of E/Ec for the two cases will be 0.3346 and 0.686, respectively.
Thus we observe that placing the SMA wires in rubber sleeves is quite detrimental to the shaft stiffness, because of the relatively larger volume occupied by the rubber.In the above example, the volume fraction of rubber sleeves Vr in the composite shaft will be three times that of Vn.Thus when Vn=2%, then Vr=6% and the volume fraction for the composite Vc=92%.If Vn--20%, then Vr=60% and Vc=20%, which obviously is not a desirable proportion.Therefore, when the shaft stiffness is an important considera- tion requiring larger values of Vn, the rubber sleeve dimensions must be either reduced drastically or the rubber sleeves should be eliminated altogether.If rubber sleeves are not used (rhn 1), the values of El Ec for two cases (unactivated and activated wires) as shown in Table I are 0.9935 and 1.0286 for Vn=2%, 0.9676 and 1.143 for Vn=10% and 0.9352 and 1.286 for Vn=20%.The aspects of fabrication of shafts with or without the rubber sleeves are beyond the scope of the present paper.

Effect of Stiffness Variation of Wires
Next, consider the shaft with NITINOL wires in rubber sleeves with Vn=2%, but without any tensile force i.e., rt 0. For rhn--4, E is 34.54 and 35.85GPa for unactivated and activated wires, respectively.Assuming mass density p of compo- site, NITINOL and rubber to be 1600, 6400 and 1200 kg/m3, respectively, the value of m from Eq. ( 7) works out to 1.182.The rotor natural frequencies for the unactivated and activated wires become 321.09and 327.12rad/s, respectively.It is obvious that the variation in natural frequency of about 1.88% from 321.09 to 327.12 rad/s is not adequate to avoid resonance in rotor coast up/down.If we increase the NITINOL volume fraction to 20% (Vn 0.2), keeping all other factors same as before, the two natural frequencies work out to 163.2 and 233.66 rad/s.This variation of natural frequencies between the unactivated and activated wires is about 43%, which will be quite adequate for a resonant free coast up/down operation.However, it is at the expense of shaft stiffness and results in decrease of rotor critical speed.For Vn= 10%, these values are 250.84 and 281.87 rad/s for unac- tivated and activated wires giving a 12.4% varia- tion.These calculations are summarised in Table I.

Combined Effect of Stiffness Variation and Wire Tension
Let us assume that the phase recovery stress in the wires is of the order of about 250MPa, which corresponds to about 3% initial strain (Rogers, 1990).The total tensile force in all the wires will be T-(r/4).(502--402) 10 -6. 250.10 6. Vn, which is (see Table II) 3.534, 17.617 and 35.34kN for Vn equal to 2%, 10% and 20%, respectively.The value of rt-T/(EI/12) for various cases is given in column 4 of Table II.The effect of tension in wires on rotor natural frequency is evident in Table II.Column 6 gives the percentage increase in rotor natural frequency due to tension in wires alone.The value of natural frequency of rotor with wires activated and effect of tension in wires accounted for, is given  for various cases in column 7. The net percentage increase in rotor natural frequency compared to the case of unactivated wires (column 9 of Table I) is given in last column ofTable II.It is to be noted that the percentage increase (20.1% to 91%) for Vn 10% and 20% is quite substantial and adequate for a safe rotor coast up/down.Comparison of results of last columns of Tables ! and II clearly shows the dominant effect of tension in wires on rotor natural frequency.For example, for the case of Vn 20% and rhn 1, the percentage increase in natural frequency rises from 17.27% to 39.57% when the effect of tension in wires is accounted for.Results for Vn-20% also show that tension in wires offsets considerably the loss of bending rigidity of the shaft.For example, the rotor natural frequency increases from 304.54 to 362.46rad/s (19.2% increase) for rhn--1 and from 233.66 to 311.82 rad/s (33.45% increase)   for rhn-4.
Combined Effects of Wire Stiffness, Wire Tension and Support Stiffness Shape memory alloy can also be used in the rotor support system (Nagaya et al., 1987).Liang and  Rogers (1993) have discussed the variation of stiffness of springs made of the shape memory material.It is shown that the stiffness of the springs, which is proportional to Young's modulus of the material, could be made to vary 3-4 times by activating the shape memory material.It will be desirable to control the rotor support stiffness by using the shape memory material in addition to the control of rotor shaft stiffness by varying the wire Young's modulus and the tension (shape memory effect).The combination of all the three factors will provide maximum flexibility to the designer to alter rotor critical speeds in a wide range.
For the case of flexible supports, two configura- tions as shown in Fig. 7 can be considered.Here Kb represents the stiffness of the pedestals or/and that provided by fluid film or rolling element bearings.
Kssp represents the stiffness of the spring made of the shape memory material which can be placed either in series (configuration I) or in parallel (configuration II).The effective support stiffness is K= Kb" Kssp/(Kb + Kssp) and (Kb + Kssp) for configurations I and II, respectively.It is obvious that for Kssp to have appreciable effect, Kb should be sufficiently rigid in configuration I and should be sufficiently flexible in configuration II.The choice of configuration I or I! therefore will depend on the relative value of Kb.Thus, configuration ! will be desirable when Kb >> Kssp while configuration I! is desirable when Kb<<Kssp.The frequency equation for the Jeffcott rotor on flexible supports is obtained from expression ( 4) by setting f =fl =f2-and rm-l" PsZs-17.6(1+ r,/r2) + 11.3543rk(1 + r,r2) 2 + r/rc 2 + 34.56r(1 + r,/2)2). ( If Vn 0, then E= Ec and r 0. Let us assume r (K Ks) when the SMA in the rotor support system is unactivated.From Eq. ( 17), z 2 P/P-0.099.Therefore p 759.7.0.099/2 239.03 rad/s.
If stiffness K increases three times because of acti- vation of SMA in rotor supports then, r= 1/3.
Table III gives for the Jeffcott rotor (=0), frequency for five cases ranging from one extreme (p) condition when both the SMA in rotor support and the wires in rotor shaft are unactivated to the other extreme (Ps) when both (the supports and the wires) are activated including the effect of tension in wires.The natural frequencies Pl to P5 in rad/s given in Tables III-VI correspond to following five cases: Pl SMA in rotor support and wires unactivated; P2 SMA in rotor support unactivated, wires in shaft activated; tension in wires present; P3 SMA in rotor support activated, wires in shaft unactivated; P4 SMA in rotor support and shaft wires activated, tension in wires absent; Ps SMA in rotor supports and shaft wires activated, tension in wires present.
In the present analysis, the value of Kssp for the unactivated and the activated supports is assumed to be 48EcI/l and 144EcI/1 , respectively.Therefore the effective value of rk in Eq. ( 17) is E/Ec for the unactivated support and E/3Ec for the activated support.Results of Table III show that substantial percentage increase (Ps-Pl)" 100/pl in natural frequency is obtained.This value is 19% for Vn=2% and 34-38% for Vn= 10% and about 50% for Vn 20%.The combination of Vn 20% and rhn 4 can be ignored because Vc for this case is only 20%.The values of P2 and p show that in 210 K. GUPTA  general the percentage increase (P2-Pl)" 100/pl in natural frequency, when only wires are activated, increases with increasing Vn.For example it is 2.7% for Vn=2% and 20.5% for Vn=20%, when rhn= 1. Percentage increase (P3-Pl)" 100/pl is around 15% for all values of Vn and rhn.The case of Vn 20% and rhn 4 is impracticable and hence ignored in further discussion.Comparison of results of last two columns of Table III clearly shows that the effect of tension in wires becomes more dominant as the value of Vn increases.For rhn 1, the percentage increase in the natural frequency due to this effect rises from 16.8 to 19.3 for Vn 2% and from 30.5 to 49.2 for Vn 20%.
For the two-mass rotor =0.5.Substituting 0.5 and rm-in Eq. ( 4), we get p2 (10.24(1 + r,/Tr2) + 11.3543rk(1 + rz/Tr2) 2) (30.72 + 62.9951rk(1 + r,/Tr2) + 34.56r (1 + r,17r2) 2). (8) Results from frequency equation ( 18) are given in Table IV.Comparison of results of Tables III and  IV shows that the general trend of results remains the same when 0.5.The rotor natural frequency in all cases has increased because of splitting of the total mass 2M in two lumped masses M each at a distance of l.The relative (percentage) increase in rotor natural frequency for the various cases has also increased slightly in comparison to the case when [ 0. Therefore a Jeffcott model can be used as a first approximation of a multimass rotor system on flexible supports to assess the volume fraction of SMA (Vn) required to achieve a prescribed separation of rotor natural frequency in unactivated and activated states.
Effect of Loss of Stiffness of Composite due to Temperature Rise Baz and Chen (1993) have experimentally estimated the loss of stiffness in a composite shaft of Teflon.The storage modulus is found to decrease progres- sively with increasing temperature.Rogers (1990) has shown a sharp drop in first mode frequency of a graphite/epoxy beam with increasing temperature.
There was no tension (phase recovery forces) in the wire.His experiments have shown that the natural frequency of the graphite/epoxy beam decreases from 20Hz to about 14Hz as the temperature increases from 70F (21C) to 150F (65.5C).This indicates reduction of stiffness of graphite/epoxy composite to about 1/2 of its value at 21C.
Assuming the same reduction in stiffness of composite (Ec 17.5 GPa), the natural frequency for five cases for the Jeffcott and the two-mass rotor on flexible supports are given in Tables V and VI respectively: It may be noted that the results of case 3 (SMA in supports activated and the wires unactivated) remain unaffected for Jeffcott rotor in Tables III and V and for two mass rotor in Tables IV and VI.
In both the Tables V and VI, we note that for small volume fraction (Vn 2%)p5 < Pl, i.e., reduction in Ec dominates over the stiffening effects due to tension in wires and increase in stiffness of rotor supports.As the volume fraction Vn increases, /95 >Pl.As in previous discussion, we shall ignore here the case of Vn 20% and rhn 4. Also for rn--10%, P2 <Pl when rhn--1 and P2 >Pl when rhn 4.This is primarily because Pl is smaller when rhn 4 due to presence of 30% rubber in the shaft.It may be noted that percentage increases fromp top3 and p2 to p5 (Tables III-VI) are due to activation of rotor supports.In general, the percentage increase in natural frequency increases with increasing Vn.
Results of Tables V and V! show that a percentage increase upto 40% is achievable, even when Ec reduces to halfits value due to rise in the temperature of the composite during activation of wires.
Comments on Coast Up/Down Strategies Figure 8 illustrates the possible coast up/down strategies.It is obvious that during coast up, the SMA (wires and/or rotor supports) will be initially in activated state and at around a speed of c*, the SMA will have to be deactivated.During coast down, the situation will be opposite, i.e., first unactivated and then activated as the rotor approaches c*.The coast up and coast down are shown by arrows in Fig. 8. Since the activation or deactivation of SMA can never be instantaneous where as the coast up or coast down may be rapid, it would be necessary to start the deactivation in case of coast up at a speed c < c* and start activa- tion in case of coast down at a speed c > c.The figure shows the coast down path of the rotor if instantaneous deactivation is achieved at c or c.
The actual coast up/down path will be in between the curves of activated and unactivated states as shown in Fig. 8.It may be noted that activation or deactivation of rotor supports will be easier and could be designed to be rapid enough in relation to the rate of coast up/down.The activation/deactivation of wires on rotating shafts is more cumbersome to achieve.Also, since the whole length of wire is embedded in composite shaft or in rubber sleeves, loss of heat and consequent deactivation is difficult to achieve rapidly.From the above considerations, it appears more attractive to introduce SMA in rotor supports.Introduction of SMA wires in the rotor shaft should be considered if a larger value of separation margin SM is desired.

CONCLUSIONS
From the present analysis and the numerical examples, the following general conclusions can be drawn: 1. Jeffcott model of a multimass rotor on flexible supports will be adequate to assess the order (in percent) of separation in rotor natural frequency that can be achieved by activating the shaft wires and/or the rotor supports.However, to estimate the actual value of natural frequency the multi mass rotor model should be used.In an 2. Introduction of SMA wires in a carbon epoxy composite shaft, makes the rotor relatively heavier because of large difference in mass density of SMA and the composite.The effective modulus E in comparison to Ec varies consider- ably depending upon Vn and rhn.The shaft stiffness is very sensitive to the value of particularly when Vn is more.The shaft loses its stiffness quite rapidly as the values of Vn and increase.It is obvious that larger values of rhn (>4) will restrict Vn in practical designs.
3. Tension in wires due to phase recovery stresses when wires are activated plays a dominant role in raising the rotor natural frequency.A large tension in wires offsets considerably the loss of shaft stiffness due to larger rhn or Vr.However, from practical standpoint a large value of rhn or Vr is not desirable.The effect of wire tension is most pronounced when the rotor supports are rigid.The effect decreases as the rotor supports become flexible.
4. Results of Tables III and IV show that consider- able variation (up to 50% or more) in natural frequency could be obtained by combined effects of activation of rotor supports, SMA wires in shaft and tension in wires.Variations of the order of 15-20% could be obtained from a single effect, i.e., activation of wires or activation of supports.In many situations where the coast up/down is not rapid, a smaller separation margin SM1 of an order of 15-20% may be quite adequate.To achieve a higher separation margin, typically more than 30%, combined activation of shaft wires and rotor supports will be necessary.
5. Comparison of results of Tables V and VI with those of Tables III and IV clearly shows that practical design of shaft is feasible in spite of the effect of loss of stiffness of composite because of rise in temperature due to activation of wires.
However, a larger volume of SMA (Vn > 10%) in the rotor is required to offset this effect.

FIGURE 5
FIGURE 5 Equivalent longitudinal modulus of composite shaft with SMA fibres.

FIGURE 8
FIGURE 8 Activation strategy for coast up/down.
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TABLE Frequencies of
Jeffcott rotor on simple supports without wire tension(T= 0) u unactivated; a activated.

TABLE II
Effect of wire tension on natural frequencies of simply supported Jeffcott rotor

TABLE V
Natural frequency of Jeffcott rotor on flexible supports with loss of stiffness of composite accounted for

TABLE VI
Natural frequency of two-mass rotor on flexible supports with loss of stiffness of composite accounted for