fatigue

Article

Dynamic analysis of shafts of compositematerials

V Alwan1, Ankit Gupta1, AS Sekhar1 and R Velmurugan2

Abstract

The objective of this study is to analyze the dynamic behavior of composite shafts with particular interest on estimation

of damping. The composite shafts have been chosen as potential candidates for the replacement of conventional metallic

shafts in many applications. This study analyzed the dynamic behavior of different materials such as glass/epoxy, car-

bon/epoxy, and boron/epoxy at different speeds. Detailed dynamic studies on the effect of different materials, stacking

sequences on the unbalance responses, and eigenfrequencies are required, particularly with respect to the damping

present in the composite material. Different methodologies such as logarithmic decay curve, half-power method, and

hysteresis loop method using force sensors have been used for determining the damping of the composite shafts.

Composite tube-shafts and solid shafts have been modeled and analyzed using ANSYS. A parametric study has been

carried out to analyze the effect of stacking sequence on eigenfrequencies of composite tube-shafts. This study aims to

have a comprehensive view of numerical and experimental analyses of composite shafts for eigenvalues, damping

estimation, and unbalance response. Eigenfrequencies of composite tube-shafts and solid shafts were estimated through

modal testing and are compared with analytical and FEM results. In this study, torsional vibration analysis for composite

solid shafts has also been carried out using the rotational laser vibrometer.

Keywords

Composite, damping, FEM, rotordynamics, shaft, torsional vibrations

Introduction

Composite materials are widely used as alternatives forconventional materials because of their desirable andtailorable properties that could not be achieved byeither of the constituent material acting alone. Theirsuperior strength-to-weight ratio, greater specific stiff-ness, and properties such as good wear resistance, longfatigue life, durability, thermal, electrical, and acousti-cal insulation, etc., have made them attractive alterna-tives for conventional materials in a wide variety ofproducts. The most common example of a compositematerial is reinforcing fibers embedded in a matrix.Fiber orientation in each layer as well as the stackingsequence of various layers can be controlled to generatea wide range of physical and mechanical properties forthe composite laminate. The fibers form the principalload carrying members, while the surrounding matrixacts as a load transfer medium between the fibers.The surrounding matrix also keeps the fibers in desiredlocation and orientation, and also protects them fromenvironmental damages due to temperature and

humidity. Rotating machinery is one of the mostimportant classes of machinery, which is extensivelyused throughout the industrial world in diverse appli-cations such as power stations, marine propulsion sys-tems, aircraft engines, machine tools, automobiles,medical equipment, household accessories, etc. Theapplication of composite shafts has come a long wayfrom early low-speed automotive drive shafts to heli-copter tail rotors operating above the second criticalspeed. With the operation of composite rotor at super-critical speeds, a substantial amount of payoffs and netsystem weight reductions are possible. Dynamicinstability, which causes violent whirling of the shaftseventually leading to catastrophic failures of rotors, is

1Department of Mechanical Engineering, IIT Madras, India.2Composite Technology Centre, Department of Aerospace Engineering,

IIT Madras, India.

Corresponding author:

AS Sekhar, Department of Mechanical Engineering, IIT Madras, Chennai

600036, India

Email: [email protected]

Journal of Reinforced Plastics

and Composites

29(22) 3364–3379

! The Author(s) 2010

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DOI: 10.1177/0731684410371404

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a major cause of concern in supercritical shafting. Thestudy of damping that significantly affects the instabil-ity thresholds is fundamental, and accurate predictionof damping in composite rotors is extremely vital forthe design and safe and reliable operation of theserotors at supercritical speeds.

Analytical and experimental studies on rotordynamic aspects of composite shaft behavior are few,and are relatively recent although more than three dec-ades ago, Zinberg and Symmonds1 described boron/epoxy composite tail rotor drive shaft for a helicopter.Bauchau2 performed a series of important investigationson composite shafts, with emphasis on the determina-tion of stiffness, strength characteristics, and stressesunder unbalance conditions. The shafts were operatedin subcritical region, and the typical rotor dynamic phe-nomenon was not studied. Zorzi and Giordano3

conducted rotor dynamic experiments on compositeshafts and compared with those of aluminum shafts.

Lim and Darlow4 conducted a series of studies oncomposite shafts. The shafts were tested under no-loadcondition. A unified balancing approach was applied tobalance the shaft up to the second critical speed.Additional works on composite shafts dealing withboth the theoretical and experimental aspects werereported by Singh and Gupta.5 Rotor dynamic formu-lation based on equivalent modulus beam theory wasdeveloped for a composite rotor with a number oflumped masses, and was supported on bearings repre-sented by eight bearing coefficients. Singh and Gupta6

further studied the effect of shear-normal coupling onrotor natural frequencies and modal damping. Bert andKim7 have formulated the problem of determination ofcritical speeds of a composite shaft, including the effectof bending–twisting coupling. The two theories,namely, the equivalent modulus beam theory and layer-wise beam theory, were compared by Singh andGupta.8 Further, they9 carried out the experimentalrotor dynamic studies on two-filament wound carbon/epoxy shafts with constant winding angles (�45�,�60�). Damping was measured with the help of a raptest on rotating and non-rotating composite shafts.

El Mahdy and Gadelrab10 carried out a comparativestudy on the free vibration characteristics between theunidirectional fiber reinforcement composite and tradi-tional material rotors. Gurban and Gupta11 analyzedthe natural frequencies of composite shafts. They usedmodified equivalent modulus beam theory, whichincludes the gyroscopic effect, shear deformation, androtary inertia. The modification includes the effects ofstacking sequence and different coupling mechanismspresent in the composite material.

Recently, Sino et al.12 included the effect of internaldamping in their simplified homogenized beam theoryto analyze the sensitivity of the frequencies and

instability thresholds regarding shear effect, stackingorder, and fiber orientation. Boukhalfa et al.13 consid-ered the p-version of the hierarchical finite elementmethod (FEM) to perform free vibration analysis ofrotating composite shafts, modeled with theTimoshenko theory.

In this study, the dynamic analysis of compositetube-shafts and solid shafts have been carried out bystudying their vibration characteristics such as naturalfrequencies both in bending as well as torsional modesand also through the estimation of damping.Experimental investigation is carried out to calculatethe flexural and torsional natural frequencies.Experimentally determined properties were used inFEM analysis to simulate the dynamic behavior ofthese composites, and parametric studies on the effectof stacking sequence and material were carried out.

Flexural vibration analysis using FEM

Composite shaft modeling

Tube-shafts. The finite element (FE) formulation forthe composite shaft is based on the first-order sheardeformation theory. The modeling details are given inSekhar and Srinivas.14 These are not discussed here, asthe presented model adopts only ANSYS. Compositetube-shafts are modeled through ANSYS with SHELL99 element, which is an eight-noded element, containssix degrees of freedom (dof) in Ux, Uy, Uz, ROTX,ROTY, and ROTZ at each node. This element allowsdifferent orientation, thickness, and material for eachindividual layer in the composite, and up to 250 layerscan be analyzed with this element. Typical FE used forthe purpose of analysis is shown in Figure 1.

Figure 1. SHELL geometry.

Alwan et al. 3365



Solid shafts. Composite solid shafts are modeledin ANSYS using BEAM 188 elements. It has twonodes and six dof per node, which include three dis-placements and three rotations. A typical element isshown in Figure 2.

Analysis of composite tube shaft

Boron/epoxy tube-shaft with the following data hasbeen considered for validation.

. Length of the shaft¼ 2470mm

. Density of the shaft material¼ 1970 kg/m3

. Mean radius of the shaft¼ 63.5mm

. Wall thickness¼ 1.321mm

. 10 layers of equal thickness from the innermost layer[90�, 45�, 45�, [0]6, and 90�]

. E11¼ 210GPa, E22¼ 24.1GPa, G12¼ 6.9GPa,t12¼ 0.36 (E, elastic modulus; G, shear modulus)

. SHELL 99 is used for the analysis in ANSYS

. For the end supports, COMBIN 14 (spring damper)elements with stiffness value of 107N/m were used.

The present FE analysis for the above system per-formed through ANSYS gives the natural frequency ofabout 90.33Hz, which is equivalent to 5420 rpm of theshaft. The corresponding values obtained throughdifferent theories for the same rotor are given inTable 1, where a good agreement of the presentedresult can be noticed.

Using the same ANSYS modeling, a parametricstudy has been carried out on a composite shaft ofthe following data:

. Length of the shaft¼ 1000mm.

. Radius of shaft¼ 50mm.

. Thickness¼ 16mm.

. Bearing stiffness¼ 100N/mm.

Composite shafts made of carbon/epoxy,boron/epoxy, and graphite/epoxy are considered forthe purpose of analysis. The properties of these mate-rials are given in Table 2. In the ANSYS modeling, atthe boundaries, the displacements are constrained andthe rotations are unconstrained. The bearings are rep-resented in ANSYS by an element called COMBIN 14,which is a spring damper element. The bearing stiffnessof the spring is 107N/m. Analysis is carried out fordifferent stacking sequence and the results are givenin Table 3. To study the effect of stacking sequence,four layers of equal thickness (4mm) with different

Table 1. Values of critical speed through different theories

Investigator Critical speed (rev/min) Method

Zinberg and Symmonds1 5500 Experimental

5780 Equivalent modulus beam theory

Singh and Gupta6 5620 Layerwise beam theory including shear effect

Gubran and Gupta11 5555 Layerwise beam theory without including Poisson’s effect

5552 Modified equivalent modulus beam theory without

including Poisson’s effect

Sino et al.12 5767 Simplified homogenized beam theory

without including shear effect

5435 Simplified homogenized beam theory including shear effect

This study 5420 Through ANSYS (with shear effect)

Figure 2. Beam element geometry.

Table 2. Properties of different composite materials

Material

E1

(GPa)

E2

(GPa) �12

G12¼G13

(GPa)

G23

(GPa)

q(kg/m3) E1/�

Boron/Epoxy 211 24.1 0.36 6.9 6.9 1967 0.107

Graphite/Epoxy 139 11 0.313 6.05 3.78 1578 0.088

Carbon/Epoxy 130 10 0.25 7.0 7.0 1500 0.087

3366 Journal of Reinforced Plastics and Composites 29(22)



stacking sequences are considered for the analysis.A typical bending mode of the composite shaft isshown in Figure 3.

Analysis is also carried out by increasing the numberof layers to six and tried for different combinations ofthe stacking sequence. In this analysis, in addition toorientations 0 and 90, layers with orientations 30, 60,45, and �45 are also tried and the effect of stackingsequence is studied. The results of the analysis aregiven in Table 4.

It is observed from Table 3 that the type of materialsand their stacking sequence have a great influence onthe natural frequency of a composite shaft. Also, it isnoticed that by keeping the innermost layer in thetransverse direction, we can increase the naturalfrequency of the system for the same material. Whencompared to different materials, for the same stackingsequence, boron/epoxy shafts exhibit maximum

eigenfrequency as the specific modulus (E/q) of theboron/epoxy is higher than the other materials.In this analysis, the ply orientation is taken to be com-prised of multiply varying 0/90 (cross ply) combination.Also, from Table 4, it is observed that boron/epoxycomposite shaft is superior to other composites,which will be useful to make stiff rotor shafts.

Analysis of composite solid shaft

Composite solid shafts of different diameters are mod-eled in ANSYS to find the natural frequencies of thefirst three modes. Parametric study has been carried outfor a simply supported rotor system having a spanlength of 88 cm between the bearings. Different diame-ters and materials have been considered and comparedwith the theoretical results. The properties of materialof composite shafts considered for the analysis aregiven in Table 5. The natural frequencies are obtainedfor steel and composite shafts with different diame-ters and the results are given in Tables 6–8. Thetheoretical results are obtained using the expressionsfrom Reddy.15

The first three natural frequencies of the glass/epoxy,carbon/epoxy, and boron/epoxy composite shafts withdifferent diameters ranging from 30 to 80mm are givenin Tables 6–8, respectively. The corresponding theoret-ical values obtained for the first three modes are alsopresented. Results show that the presented results

Table 3. Effect of stacking sequences and materials on the

eigenfrequencies

Eigenfrequencies (Hz)

Type of material

Stacking

sequence Mode I Mode II Mode III

Carbon/Epoxy 0/0/90/90 382 990 1683

90/90/0/0 410 1043 1832

90/0/90/0 415 1053 1769

Boron/Epoxy 0/0/90/90 388 969 1603

90/90/0/0 437 1107 1823

90/0/90/0 433 1085 1793

Graphite/Epoxy 0/0/90/90 379 976 1653

90/90/0/0 432 1134 1815

90/0/90/0 431 1111 1880

Figure 3. Typical bending mode shape of composite tube-shaft.

Table 4. Effect of different layers and materials on first

eigenfrequency

First eigenfrequency (Hz)

No. of

layers

Stacking

sequence

Boron/

Epoxy

Carbon/

Epoxy

Graphite/

Epoxy

4 0/0/90/90 421 382 377

0/30/60/90 439 393 387

0/45/�45/90 497 433 428

6 0/90/0/90/0/90 463 423 406

0/45/90/90/45/0 515 497 467

45/�45/45/�45/45/�45 396 379 374

Table 5. Material properties of different materials used for

analysis

Material E (GPa) t12 Density (kg/m3) E/q

Glass/Epoxy 40 0.28 1800 0.022

Carbon/Epoxy 180 0.25 1500 0.120

Boron/Epoxy 206 0.36 1950 0.106

Alwan et al. 3367



obtained from FE models agree well with the theoret-ical results. It is observed that as expected, the increasein diameter of the shaft increases the natural frequencyof the shaft in all the materials. Although boron/epoxy

shafts have exhibited a higher natural frequency in theprevious cases, here carbon/epoxy shafts are stiffer,as their E/q value is higher. We could infer fromTables 6–8 that the E/q value of carbon/epoxy(Table 5) shaft can be tuned to obtain stiffer character-istics with lesser cost than that of boron/epoxy shaft,hence making it economical.

The variation of first natural frequency with thediameter of shafts for different materials is shown inFigure 4. It can be noticed that the size of the shafthas less influence in the case of glass/epoxy, as com-pared to other two cases. A similar trend has beenobserved in the case of higher modes. Figure 5 showsthe variation in different modes in the case of car-bon/epoxy shaft. The influence of diameter is quitehigh in the case of the third mode as compared to theother modes. Similar trends are observed in the case ofother two materials.

Unbalance response of composite solid shafts

Composite solid shafts made of glass and carbon/epoxyhaving a steel disc at the center are analyzed using FEMto study the unbalance response of the system devel-oped due to the eccentricity between the disc and shafts.The length of the shaft is 1000mm and diameter 30mm.The shaft is simply supported on two end bearings.Eccentricity of 0.02mm is used in the calculation ofunbalance response. The mass of the steel disc consid-ered is 3.5 kg. Along with the composite shafts, a steelshaft is also considered and analyzed. Table 9 shows theproperties of the steel and composite solid shafts used.In this analysis, damping of 0.7%, 1%, and 2%16,17 areused for the steel, glass and carbon/epoxy shaft sys-tems, respectively.

Harmonic analysis is performed in ANSYS to findthe unbalance response. BEAM element 188 is used andthe discs are represented by the lumped masses. Theunbalance response of the steel and composite solidshafts with steel disc is shown in Figure 6.

Table 10 gives the values of peak unbalanceresponses and the corresponding speed for the steeland composite shafts mounted with steel disc. FromFigure 6, it is seen that the critical speed of the glass/epoxy shaft is less than that of the carbon/epoxy shaft.It is also observed that the unbalance response of theglass/epoxy composite shaft is also lower than that ofthe other two shafts. The noticeable difference in peakresponse between the composite shafts and steel isobserved. The critical speed of carbon/epoxy is higherthan that of steel and glass/epoxy shafts, while that ofglass/epoxy shaft is lower. This is due to higher E/qvalues as explained earlier. Since damping of the com-posite shafts is higher, the peak unbalance responsesare lower as compared to that of the steel shaft.

Table 6. Comparison of natural frequencies of glass/epoxy

shaft with different diameters

Diameter

(mm)

Natural frequency (Hz)

ANSYS Theory

Mode I Mode II Mode III Mode I Mode II Mode III

30 66.37 265.2 595.64 66.33 265.32 596.97

40 88.47 353.21 792 88.97 355.8 800.7

50 109.33 436.06 976.44 110.65 442.6 995.8

60 132.62 528.20 1180.80 132.73 530.92 1194.00

70 154.62 614.91 1370.40 154.96 619.80 1394.60

80 176.60 701.00 1558.00 176.98 707.90 1592.80

Table 7. Comparison of natural frequencies of carbon/epoxy


Diameter

(mm)


ANSYS Theory


30 166.00 665.00 1495.00 166.42 665.68 1497.78

40 222.06 886.00 1988.00 222.44 889.76 2001.90

50 274.00 1094.00 2450.00 277.70 1110.80 2499.00

60 332.00 1325.00 2962.00 333.17 1332.60 2998.53

70 388.00 1543.00 3439.00 388.95 1555.80 3500.55

80 443.00 1760.00 3910.00 444.22 1776.80 3997.98

Table 8. Comparison of natural frequencies of boron/epoxy


Diameter

(mm)


ANSYS Theory


30 156.31 624.55 1402.70 156.14 624.56 1405.26

40 208.35 831.80 1865.70 208.70 834.80 1878.30

50 257.40 1026.90 2299.50 260.59 1042.66 2345.30

60 312.00 1243.00 2779.00 312.60 1250.40 2813.40

70 364.00 1448.00 3227.00 364.94 1459.70 3284.46

80 415.00 1651.00 3669.00 416.79 1667.16 3751.10




Experimental investigation on flexuralvibration of composite shafts

Fabrication of composite shafts

Composite tube-shafts. To design tube-shafts of dif-ferent diameter, correspondingly wooden molds of dif-ferent diameter are taken and the WRM glass fibermats, which are wetted with the resin, are wrappedover the mold to get the required thickness. Thenumber of layers are varied from 5 to 12 to get thick-ness values between 3 and 7mm. The tube-shafts areallowed to cool in the room temperature and then theyare kept in the oven for 12 h at 80�C for proper curing.Finally, the tube-shafts are trimmed and machinedproperly to get uniform diameter.

Composite solid shafts. Composite solid shafts of 25and 30mm diameter are prepared through hand layuptechnique. For the fabrication of composite shafts,composite molds are made in the form of a half cylin-drical tube. Continuous unidirectional glass fibers of

500

375

250

125

020 45 70

Diameter (mm)

95

Glass/Epoxy

Carbon/Eposy

Boron/Epoxy

Nat

ural

freq

uenc

y (H

z)

Figure 4. Variation of first natural frequency with diameter for different shaft materials.

4000

3000

Nat

ural

freq

uenc

y (H

z)

2000

1000

020 45

Diameter (mm)

70 95

Mode I

Mode II

Mode III

Figure 5. Variation of different modes with diameter of carbon/epoxy composite shaft.

Table 9. Properties of solid shafts

S. No. Material

Elastic

modulus, E

(GPa)

Poisson’s

ratio

Density

(kg/m3) E/q

1 Glass/Epoxy 40 0.28 1800 0.022

2 Carbon/Epoxy 130 0.25 1500 0.086

3 Steel 210 0.3 7800 0.029

Alwan et al. 3369



required weight are taken and wetted with equalamount of resin by weight. The wetted fiber rovingsare filled in the cavity of the molds and the twohalves are joined together. The joint is tightened withthe help of bolts, which are distributed over the length.

Analysis of composite tube-shaft

Estimation of natural frequency by experiment. Thenatural frequencies of the composite tube-shafts madeof glass/epoxy are obtained by using the modal analy-sis. In the experimental setup, the tube-shafts aremounted on the three-jaw reversible chucks. The jawsare mounted on the plumber blocks. The bearings areseated on the plummer block supporting the three-jawchucks. The following instrumentation has been usedfor this experimental study:

1. Impact hammer: Kistler 500 lbf capacity, sensitivity50mv/lbf.

2. Displacement pickup: Eichhorn + hansmann,0–1mm, sensitivity 10V/mm.

3. Amplifier: Eichhorn + hansmann MW 210-13-3,50mV/cm.

The steel ring is placed on the tube-shaft for the sakeof measurements using non-contact probes. The fre-quency response plots are obtained for differentimpacts at various locations. Response from the capac-itance pickup is amplified by the amplifier and theseresponses are fed to the RT Pro software through thedata acquisition system (DAS), which in turn generatesthe frequency response function (FRF) plots fromwhich the different natural frequency values areobtained. The experimental results of the natural fre-quencies of glass/epoxy tube-shafts obtained fordifferent diameters and thickness values are givenin Tables 11 and 12.

Estimation of natural frequency by FE analysis(ANSYS). The composite tube shafts are modeled inANSYS as explained in ‘Analysis of composite tube-shaft’. The natural frequencies are obtained for tube-shafts of diameters 99 and 60mm for different thicknessvalues and are compared in Tables 11 and 12. From thecomparison it can be observed that the results agree tosome extent. The effect of chucks and some experimen-tal errors could be the reasons for deviations.

Analysis of composite solid shaft

Estimation of natural frequency by experiment. Thenatural frequencies of the composite solid shafts madeof glass/epoxy are obtained by the modal analysis

350.00

280.00

210.00

Glass shaft

Steel

Carbon shaft

Am

plitu

de (

mm)

140.00

70.00

0.00

0 900 1800 2700

Speed (rpm)

3600 4500

Figure 6. Unbalance response of solid shafts with steel disc.

Table 10. Peak unbalance response and speed of the solid

shafts with steel disc

Type of

shaft

Peak

response (mm)

Speed at peak

response (rpm)

Glass/Epoxy composite 133 1260

Carbon/Epoxy composite 171 2835

Steel 313 2415




technique. In this technique, the specimens are sup-ported at the ends with the help of bearings andslight impact load is applied on the shaft. The positionis also varied along the span and the impulse responsewas acquired. The accelerometer and impact hammerare used to collect the response of the shaft, and thesevalues are fed to the RT Pro software through the DASto get the frequency values. Using FFT/FRF (FFT, fastFourier transform) of the data, the natural frequenciesof different modes are obtained. The frequency valuesobtained for the composite shafts of diameters 25 and30mm and for the steel shaft of diameter 28mm aregiven in Table 13. A typical FRF is shown in Figure 7.

Estimation of natural frequency by FE analysis(ANSYS). The composite and steel shafts are modeledusing FE and analyzed through the ANSYS (BEAM188 element). The comparison between first three nat-ural frequency values obtained from the experiments

and the values obtained through ANSYS are given inTable 14, and it can be observed that the results agreefairly well.

Unbalance response of composite shafts

Unbalance response is carried on for the compositesolid shaft made of glass/epoxy having steel disc atthe center. The unbalance response of shaft for anyconstant speed was measured with help of laser vibrom-eter (RLV 5500), which focuses on the laser beam onthe reflecting tape wrapped around the disc. The signalfrom the laser vibrometer was sent to the DAS, anddata analysis has been carried out in the Dewetron soft-ware to obtain the unbalance response for differentspeeds. The diameter of the shaft used for the experi-ment is 30mm and length 900mm. The shaft is modeledin ANSYS using BEAM 188 element whose dampingvalue is assumed as given earlier in ‘Unbalanceresponse of composite solid shafts’ section and theunbalance response has been plotted and comparedwith experimental result as shown in Figure 8. A shiftin peak was observed in critical speed obtained fromthe experiment for glass/epoxy solid due to slight flex-ibility of actual bearings as compared to the assumedrigid boundary conditions in ANSYS.

40

30

20

10

0

–10–30 20 70

Frequency (Hz)

Frequency response function

dB (

gn/N

ewto

n)

120 170 220 270

–20

–30

Figure 7. Typical FRF plot for composite shaft of diameter 30 mm.

Table 13. Experimental values of steel and composite shafts


S. No. Diameter Mode I Mode II Mode III

1 28 mm, steel 74.4 280 575

2 25 mm, composite 32 103 247

3 30 mm, composite 38 123 260

Table 11. First natural frequency (Hz) of composite tube-shaft

of diameter 60 mm

S. No. Type Thickness (mm) Experimental ANSYS

1 8 layer 5 110 115

2 5 layer 3 55 70

Table 12. First natural frequency (Hz) of composite tube-shaft

of diameter 99 mm

S. No. Type Thickness (mm) Experimental ANSYS

1 8 layer 5 142 206

2 5 layer 3 125 135

Alwan et al. 3371



Estimation of damping of composite shafts

The damping values of composite tube-shafts and solidshafts systems are obtained for static and dynamiccases. The damping factors for static cases are obtainedby modal testing and logarithmic decrement tech-niques. The damping factors for dynamic cases areobtained by hysteresis loop technique. The shafts arerun at different speeds and the hysteresis loopsare obtained at these speeds.

Logarithmic decrement method. This method givesthe damping factor at fundamental natural frequencyfor under-damped system. Figure 9 shows a typicalamplitude decay curve for an under-damped system.The expression to calculate the damping factor (z) isgiven by:

� ¼�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4�2 þ �2p

� �, ð1Þ

� ¼1

nlog

X1

Xnþ1

� �, ð2Þ

where � is the logarithmic decrement of responsebetween n cycles.

Half-power point method. A typical response curveshown in Figure 10 can be used for estimating dampingat any eigenfrequency.

The expression for finding the damping ratio is givenby:

� ¼�!

2!n, ð3Þ

where �x is the bandwidth at the half-power points ofresonant peak for n-th mode and xn the resonantfrequency.

Hysteresis loop method. This method has been usedfor estimating the damping of rotor at any rotatingspeed. Elliptical response curve will be obtained by

1800

1600

1400

1200

1000

800

600

400

200

00 500 1000 1500 2000 2500

Speed (rpm)

ExperimentalANSYS

Am

plitu

de (

mm)

Figure 8. Unbalance response plot for the glass/epoxy solid shaft of 30 mm.

Table 14. Comparison of natural frequencies of composite solid shafts

Mode I Mode II Mode III

S. No. Diameter Experimental ANSYS Experimental ANSYS Experimental ANSYS

1 28 mm, steel 74.4 77.5 280 295 575 646

2 30 mm, composite 32.0 35.8 108 140 247 300

3 25 mm, composite 38.0 43.1 123 169 260 305




plotting displacement and force signals. Figure 11shows an example plot to estimate damping.Equivalent damping of the shaft at a particular speedcan be calculated by using Equation (4):

�W ¼ �hX2

� ¼h

2k, ð4Þ

where �W is the area of loop, h the hysteretic constant,X the maximum displacement in the loop along the

X-direction, and k the slope of the loop as shown inFigure 11.

Estimation of damping for the static case

Damping estimation through half-power point andlogarithmic decrement method. Glass/epoxy tube-shafts of diameters 60 and 99mm of 5, 8, and 12layers, solid shafts of diameter 30mm, and steel shaftof diameter 28mm are used for this experiment. Thefollowing instrumentation has been used for this exper-imental study.

1. Voltage amplifier: Piezo actuator drive amplifier,range 200V rms.

2. Oscilloscope: Two-channel digital phosphor,40MHz and 500MS/s.

3. Impact hammer: Kistler 500 lbf capacity, sensitivity50mv/lbf.

The experimental procedure follows the same pat-tern as given in ‘Analysis of composite solid shaft’ sec-tion and obtained typical FRF plots are shown inFigure 8. To ascertain the damping factor from theobtained FRF plots through half-power bandwidthmethod, Equation (3) is used. Logarithmic decrementcurve is obtained from the displacement sensor, whichis kept close to the tube shaft. Logarithmic decrementcurve has been obtained for each tube shaft and thedamping values from the curves are obtained usingEquation (2). The values have been compared forboth logarithmic decrement and half-power pointmethods. The results obtained for the tube-shafts aregiven in Table 15 and that of the solid and steel shaftsare given in Table 16. From the table, though no cleartrend is observed in damping values, it could beinferred that the tube-shaft of smaller diametersappears to have slightly more damping. It can beobserved from Table 16 that the damping of compositeshafts is higher than that of steel shafts, as expected.

K

X

Displacement (m)

For

ce (

N)

Figure 11. Hysteresis loop method.

Excitation frequency

P1

Amax

Dw

P2

P1 and P2 halfpower bandwidthpointsAmax

√2

Figure 10. Half-power bandwidth method to calculate

damping.

Table 15. Damping values for composite tube-shafts of differ-

ent diameter and thickness

S.

No.

Diameter

(mm)

No. of

layers

Thickness

(mm)

Logarithmic

decrement

damping (%)

FRF response

damping (%)

1 60 5 3 2.53 2.60

2 8 5 6.73 5.05

3 12 7 4.45 3.18

4 99 5 3 1.60 3.47

5 8 5 1.30 0.86

6 12 7 1.55 1.31

Xn+ 1

Time

Am

plitu

de

X1

Figure 9. Procedure to calculate damping.

Alwan et al. 3373



Estimation of damping for the rotating case

Damping estimation through hysteresis loopmethod. The shafts are run at different speeds andthe corresponding forces, acceleration, and displace-ments are measured with the help of force sensors,accelerometer, and laser vibrometer, respectively. DCmotor, which can run up to a maximum frequency of6000 rpm, has been used with variable control unit torun the motor at different speeds. After acquiring thedata from the DAS, signals were filtered to removenoise and other undesirable effects. Finally, hysteresisloops were obtained between the force and displace-ment signals for different speeds. Damping has beenobtained from the hysteresis loop and damping varia-tion has been studied with variation of speed for shaftsof different diameters. The following instrumentationhas been used for this experimental study

1. Force transducer: Kistler 9712B, 50 lbf capacity, sen-sitivity 100mv/lbf.

2. RLV 5500 Polytech laser vibrometer.

3. 4-Channel NI DAS: Hi-speed USB carrier NI-USB-9162.

4. Accelerometer: Dytran 3145A, sensitivity 100mv/g,50 g range.

The laser vibrometer was placed on ground with thehelp of a tripod, and laser beam was focused on themiddle portion of the shaft to acquire excited displace-ment. Force transducer is placed on the metallic barand pressed against with the metal strip through boltwhich is fixed on the plummer block. Displacement andforce signals were acquired and measured through theDAS and processed in the post-analysis software (Exceland Origin Pro.). Obtained displacement and force sig-nals were filtered and the plot between them gives aforce–displacement curve, which is in the form of thehysteresis loop. From the hysteresis loop, the dampingfactor has been estimated as explained earlier usingEquation (4). The estimated damping factor is plottedover the shaft speed as shown in Figure 12. It isobserved that the damping factor increases with thespeed and becomes maximum at the critical zone.

Torsional vibrations of composite shafts

Torsional and bending vibrations are important inrotating shaft systems. Typical torsional vibrationproblem occurs in automotive, marine, propulsion,and industrial applications, which includes oscillationof crank shaft, drive shaft, and component wear ingearboxes. In these cases, shaft cracking, coupling dete-rioration, and gear failure will occur in the shafts.Bending vibrations play a vital role in rotating machin-ery (reciprocating engine, turbine blade, and crankshaft). Low-order torsional vibrations are of concern

9

8

7

6

5

Dam

ping

fact

or (

%)

4

3

2

1

00 1000 2000 3000

Speed (rpm)

4000 5000

Figure 12. Variation of damping factor with speed.

Table 16. Damping values for metallic and composite solid

shafts

Logarithmic

decrement method

Half-power

method

Type

Diameter

(mm)

Average

damping (%)

Average

damping (%)

Steel shaft 28 mm 0.520 1.5

Composite shaft 30 mm 3.875 3.3




in engine reliability and high-order vibrations are influ-ential on power plant noise. So, it is vital to study thetorsional behavior of rotors in practical applicationswhile we focus on bending vibrations of the system. Anumber of situations which include unbalance, misa-lignment, bearing faults, rubs, and the effect of gear,can give rise to shaft bending (or more generally angu-lar, lateral) vibrations. Besides this, stresses in compos-ite shafts subjected to unbalance excitation andtransmitted torque are also a major concern for theresearcher to determine the maximum allowabletorque to avoid delamination in the layered shaft.However, quite a few papers are published on theissue of torsional vibrations.

This section focuses on estimation of torsional nat-ural frequencies for composite solid-shaft and tube-shaft systems. Experimental setup has been developedand the obtained results are compared with numericaland analytical results.

Parametric study of composite tube-shaft

FE analysis has been carried out for estimating tor-sional natural frequencies apart from bending naturalfrequencies of composite tube-shafts of different plyorientations and different materials such as glass/epoxy, carbon/epoxy, and boron/epoxy. As explainedearlier in the section ‘Flexural vibration analysis usingFEM,’ the analysis has been carried out in ANSYS 10by using SHELL 99 Element. Composite tube-shaftmade of five layers and simply supported on end bear-ings has been considered for the purpose of analysis.The properties of material used for this analysis aregiven in Table 17. The dimensions of the shaft usedfor modeling are given as follows:

1. Length of the shaft¼ 1000mm; Diameter of theshaft¼ 100mm.

2. Total thickness of the shaft¼ 3.7mm; Number oflayers¼ 5.

3. Thickness of each layer¼ 0.74mm.

FE analysis has been done to obtain first three fun-damental modes of the tube-shaft for three differenttypes of material and different stacking sequences.

A typical mode shape obtained through analysis isshown in Figure 13. Fundamental natural frequenciesin bending mode as well as the torsion mode for differ-ent stacking sequences, for all the three materials, aregiven in Table 18. Variations of torsional natural fre-quencies for the first three modes with different stackingsequences for all the three materials are compared andare given in Table 19.

It has been observed from Tables 18 and 19 that asthe ply of 45� is included, it increases the torsional fre-quency of the structure drastically while it decreases thebending natural frequencies. It is also observed that inlamination schemes that lower the inner layer angle,higher are the torsional frequencies. It implies thatwhen more number of layers are kept parallel to theaxis of the shaft, it increases the torsional natural fre-quency. Although the shafts with boron/epoxy are stif-fer in both bending and torsional modes, the shafts ofcarbon/epoxy are not far behind. This is a useful pointfrom the cost point of view.

Basic experimental setup for estimating torsionalnatural frequencies

A preliminary setup for finding the technique to esti-mate the torsional modes of the glass/epoxy compositeshaft has been developed. Steel circular shaft of 20mmdiameter and 300mm length has been considered forthis analysis. Torsion refers to the twisting of a shaftloaded by torque, or twisting couples. Twisting momentwas created by giving an impact to any of the two adja-cent bolts at the free end of the shaft, which in turngenerates torsional modes. The details of the shaftarrangement considered for this analysis is shown inFigure 14 where the shaft is subjected to torsionalvibration by giving an impact to the bolts by theimpact hammer to the bolts which are attached at thefree end of the shaft while the other end is fixed.

Table 17. Properties of composite tube-shaft

Material

E1

(GPa)

E2¼ E3

(GPa)

G12¼G13

(GPa) t12

Density

(kg/m3)

Glass/Epoxy 40 5 4 0.28 1800

Carbon/Epoxy 130 10 7 0.25 1500

Boron/Epoxy 206 21 6.9 0.36 1950 Figure 13. Typical III mode shape (torsional) of composite

tube-shaft.

Alwan et al. 3375



The response of the shaft is captured with the help oflaser beams from RLV 5500 laser vibrometer, by usingsingle beam and two beam methods. The measureddata are analyzed in the DAS. Figure 15(a) shows theline diagram of the experimental setup explainingthe working principle of the torsional vibration.Figures 15(b) and (c) show the graphical representationof the single laser beam and two laser beam measure-ment, respectively. Single beam and two beam methodshave been used to identify the torsional modes fromthe obtained modes.

Upon impact, the cantilever shaft is excited into allvibrational modes. Single laser beam can capture onlybending modes generated by the impact force, whereastwo laser beams can sense torsional and coupled modes,

if any. However, even in the case of the two laserbeams, some bending modes are captured. But, onlyin case of the single laser beam, torsional modes arenot possibly sensed. The source head was controlledby the RLV (rotational laser vibrometer) controller.The outputs of the controller and impact hammer areconnected to the DAS. The output of the DAS goes tothe RT Pro Data Analysis Software which is used toacquire FRF plots to estimate torsional natural fre-quency. The data from both methods are comparedto identify the torsional modes, and the results, afterchecking the repeatability, are given in Table 20.Comparison of single beam and two beam resultsyield the results for torsional modes. The torsional nat-ural frequency of the steel shafts is also estimated by

Table 18. Fundamental bending and torsional natural frequency for different ply orientation of composite tube-shafts

Fundamental natural frequency (Hz)

Glass/Epoxy Carbon/Epoxy Boron/Epoxy

S. No. Orientation Bending Torsion Bending Torsion Bending Torsion

1 [0/90/0/90/0] 236 671 404 1002 404 883

2 [90/0/90/0/90] 256 647 438 975 491 994

3 [45/�45/45/�45/45] 231 899 379 1633 380 1778

4 [90/45/90/45/90] 271 667 481 1084 522 1177

5 [0/45/0/45/0] 183 676 293 1099 328 1191

6 [45/0/45/0/45] 189 1056 301 1886 334 2036

7 [30/60/30/60/30] 200 609 328 957 371 1119

8 [0/60/0/60/0] 205 674 337 1028 359 1038

9 [90/60/90/60/90] 276 643 410 734 426 748

10 [60/30/60/30/60] 210 599 344 939 384 1090

Table 19. First three torsional natural frequency values for different ply orientation of composite tube-shafts

Torsional natural frequencies (Hz)

Glass/Epoxy Carbon/Epoxy Boron/Epoxy

S. No. Orientation I II III I II III I II III

1 [0/90/0/90/0] 671 1342 2014 1002 2005 3007 889 2026 3038

2 [90/0/90/0/90] 647 291 1922 975 1946 2908 994 1923 2975

3 [45/�45/45/�45/45] 899 1818 2825 1633 3312 4920 1778 3659 5431

4 [90/45/90/45/90] 667 1330 2002 1084 2155 3311 1177 2344 3403

5 [0/45/0/45/0] 676 1356 2042 1099 2104 3165 1191 2307 3470

6 [45/0/45/0/45] 1056 2121 3200 1886 3773 5690 2036 4068 6122

7 [30/60/30/60/30] 609 1222 1836 957 1918 2879 1117 2240 3470

8 [0/60/0/60/0] 674 1350 2029 1028 2058 3092 1038 2077 3120

9 [90/60/90/60/90] 643 1286 1193 734 1482 2087 748 1449 2158

10 [60/30/60/30/90] 599 1190 1795 939 1878 2809 1089 2189 3270




standard formulae18 and the results of experiments arecompared with the analytical results in Table 20. Theexperimental results show a good agreement with thetheoretical results. This technique has been used forcomposite shafts as discussed in the section ‘Torsionalnatural frequency of composite solid shaft’.

Torsional natural frequency of composite solid shaft

The technique explained in the section ‘Basic experi-mental setup for estimating torsional natural frequen-cies’ has been used to find the torsional modes forcomposite solid shaft system, the schematic representa-tion of which is shown in Figure 16. The arrangement isdone by fixing one end with bearing (right-hand side)and fixing the small steel ring that has symmetricallylocated three bolts around the periphery, on the other

side of bearing. The ring has been purposely kept closeto the bearing to avoid the bending modes.

The composite shaft of 20mm diameter and length880mm is fixed at one end and is subjected to torsionalload at the other end. To apply the torsional load, adisc is attached to the free end which has three bolts oflarger length. These bolts are subjected to impactthrough an impact hammer which in turn produces tor-sional vibration of the shaft.

The torsional movement of the shaft is measuredwith the help of a laser beam. A reflection sticker ispasted on one of the bolts on which the laser beamisfocused. The torsional oscillation of the shaft is cap-tured with the help of the laser beam, which is reflectedby the reflecting tape. The laser beams are generated byan RLV source head and controlled by the RLV

Table 20. First torsional natural frequency (Hz) of steel shaft

Theory Experiments

S. No.

Bending

(Hz)

Torsion

(Hz)

Single laser

beam (Hz)

Two laser

beam (Hz)

1 153 2610 126 126

2 988 7830 786 781

3 2656 13,051 1552 1552

4 – 18,272 1962 2094

5 – – 2640 2607

The bold values show a closer agreement of first torsional mode between

theory and experiment.

(a)

Impact hammerDAS

RLV controller RT Pro

Shaft

Sensor head

Impact ImpactSingle laser beam Two laser beam(c)(b)

RLV sourcehead

Zoomed view infigure 18(b) and (C)

Figure 15. Schematic representation of experimental setup for 20 mm steel shaft: (a) line diagram showing the experimental setup;

(b) single beam measurement; and (c) two beam measurement.

Impact hammer

300 mn

Figure 14. Details of shaft arrangement.

Alwan et al. 3377



controller. The response of the impact hammer passesthrough conditioner and the laser beam reflections areconnected to four-channel NI DAS. Signals from theDAS are analyzed in the Dewesoft software whichprovides the FRF plots, which give the informationof the torsional natural frequency of the shaft. The tor-sional natural frequencies are obtained for the firstthree modes of the composite shaft and are comparedwith analytical results. The results are given inTable 21.

Comparison of results

The experimental composite shaft of diameter 22mmand length 880mm is modeled in ANSYS for findingthe torsional natural frequencies. SOLID 45 element isused for this purpose. This element has three degrees offreedom with eight nodes in each element. The numberof elements used for the analysis is 344. Modal analysisis carried out to find the natural frequencies of theshaft. The torsional modes are visually identified andthen extracted. The torsional natural frequency valuesthus obtained for the first three torsional modes aregiven in Table 21 and are compared with FEM(ANSYS) and theoretical18 results. It can be noticedthat the results agree fairly well.

Conclusion

Parametric study has been done for composite tube-shafts and solid shafts, and the effects of stackingsequences and materials on eigenfrequency have beenanalyzed. This can be useful for designing the compos-ite shafts for different requirements. It has beenobserved that inclusion of layer along the transversedirection into boron/epoxy material is a good optionfor manufacturing stiffer shafts. Parametric study hasalso been done for free vibration characteristics of uni-directional composite solid and metallic shafts with

steel disc. It has been observed that glass/epoxy com-posite shaft exhibits lowest eigenfrequency as a shaftmaterial. Besides this, study of unbalance responses ofmetallic and composite solid shafts with steel disc hasbeen carried out in ANSYS which can be quite usefulfor giving an option to choose for different speed oper-ations to get minimum unbalance. Parametric study fordifferent orientations and materials has been carriedout which shows that inclusion of 45� layer in the lam-inate scheme and keeping that as the innermost layer instacking sequence increases torsional frequency drasti-cally. Although the shafts with boron/epoxy are stifferin both bending and torsional modes, the shafts ofcarbon/epoxy are not far behind if tuned with betterspecific elastic modulus values (E/q). This will bequite useful from cost point of view.

Modal analysis on composite tube shafts and solidshafts has been performed and experimental resultsagree fairly well with the ANSYS results. The compos-ite shaft rotor system damping has been estimatedexperimentally by three different methods. Hysteresisloop method has been demonstrated for estimatingthe damping at any speed which can be quite a valu-able tool for estimating the damping at differentspeeds. Experimental evaluation of torsional naturalfrequency has been demonstrated in this study.The results agree fairly well with analytical andtheoretical results.

Table 21. Torsional natural frequencies of composite shaft


Mode Experiment ANSYS Theory

I 355 368 373

II 1075 1103 1120

III 1850 1838 1866

Impact hammer

Shaft

Steel ringSensor head

Bearing supports

Conditioner NI card

RLV controller PCRLV sourcehead

Figure 16. Schematic representation of experimental setup for estimating torsional natural frequency of composite shaft.




Funding

The financial support received from the Aeronautics R&D

Board-structures panel through the project DARO/08/1051332/M/I is greatly acknowledged.

References

1. Zinberg H and Symmonds MF. The development of an

advanced composite tail rotor drive shaft. In:

Proceedings of 26th Annual National Forum of American

Helicopter Society, Washington, DC, 1970, pp.1–4.

2. Bauchau OA. Design, manufacturing and testing of high

speed rotating graphite epoxy shafts, PhD Thesis,

Department of Aeronautics and Astronautics, MIT,

USA, 1981.

3. Zorzi ES and Giordano JC. Composite shaft rotor

dynamic evaluation. In: ASME Design Engineering

Conference on Mechanical Vibration and Noise, Cincinnati,

Ohio, ASME Paper No. 85-DET-114, 1985, pp.1–8.4. Lim JW and Darlow MS. Optimal sizing of composite

power transmission shafting. J Am Helicopter Soc 1986;

31: 75–83.5. Singh SP and Gupta K. Dynamics of composite rotors. In:

Indo US Symposium on Emerging Trends in Vibration and

Noise Engineering, IIT New Delhi, 1996, pp.48–69.6. Singh SP and Gupta K. Composite shaft rotor dynamic

analysis using a layerwise theory. J Sound Vib 1996; 191:

739–756.

7. Bert CW and Kim CD. Dynamic instability of composite-

material drive shaft subject to fluctuating torque and/or

rotational speed. Dyn Syst 1995; 10: 125–147.

8. Singh SP and Gupta K. Dynamic analysis of composite

rotors. Int J Rotating Mach 1994; 2: 179–186.

9. Singh SP and Gupta K. Damping measurements in

fibre composite rotor. J Sound Vib 1998; 211:

513–520.

10. El Mahdy H and Gadelrab RM. Free vibration of unidi-

rectional fiber reinforcement composite rotor. J Sound

Vib 2000; 230: 195–202.

11. Gubran HBH and Gupta K. The effect of stacking

sequence and coupling mechanisms on the natural

frequencies of composite. J Sound Vib 2005; 282:

231–248.12. Sino R, Baranger TN, Chatelet E and Jacquet G.

Dynamic analysis of a rotating composite shaft.

Compos Sci Technol 2007; 68: 337–345.13. Boukhalfa A, Hadjoui A and Hamza Cherif SM. Free

vibration analysis of a rotating composite shaft using

the p-version of the finite element method. Int J

Rotating Mach 2008; 1: 1–10.14. Sekhar AS and Srinivas BN. Dynamics of cracked com-

posite shafts. J Reinf Plast Compos 2003; 22: 637–653.

15. Reddy JN. Mechanics of laminated composite plates:

Theory and analysis. Boca Raton: CRC Press, 1997,

p.217.

16. Ankit Gupta, Velmurugan R, Sekhar AS and Alwan V.

Damping characteristics of glass epoxy composites sub-

jected to thermal loading. RSM Int J Eng Technol

Manage ISSN: 0974–9535 (in print).17. Alwan V, Sekhar AS and Velmurugan R. Mechanical

characterization and DMA studies of carbon/epoxy com-

posites. In: Proceedings of ISAMPE National Conference

on Composites (INCCOM-8), Thiruvananthapuram,

4–5 December, 2009, p.376.

18. Rao SS. Mechanical vibrations. NY: Addison–Wesley

Publishing Company, 2000.

Alwan et al. 3379



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