14
Journal of Sound and Vibration (I 978) 56(2), 187-200 FINITE ELEMENT EIGENVALUE ANALYSIS OF TAPERED AND TWISTED TIMOSHENKO BEAMS R. S. GUPTA Department of Mechanical Engineering, Punjab Engineering College, Chandigarh-11, India AND S. S. RAO Department of Mechanical Engineering, Indian Institute of Technology, Kanpur-208016, India (Received 13 January 1977, and in revisedform 14 September 1977) The stiffnessand mass matrices of a twisted beam element with linearly varying breadth and depth are derived. The angle of twist is assumed to vary linearly along the length of the beam. The effects of shear deformation and (otary inertia are considered in deriving the elemental matrices. The first four natural frequencies and mode shapes are calculated for cantilever beams of various depth and breadth taper ratios at different angles of twist. The results are compared with those available in the literature. 1. INTRODUCTION The analysis of tapered and twisted beams has wide application in many industrial problems. The vibration and deflection analysis of compressor blades, turbine blades, aircraft pro- peller 131ades, helicopter rotor blades, gear teeth, springs of electromechanical devices, electrical contact switches, etc., all can be made by using such beam elements. Tapered beams have been analyzed by many investigators using different techniques. Rao [1] used the Galerkin method to calculate the fundamental natural frequencies of beams tapered in depth. Housner and Keightley [2] applied the Myklestad [3] procedure to determine the first three modes of a tapered beam. Martin [4] obtained the frequencies of a tapered beam using the assumption that the eigenvalues and eigenvectors of a tapered beam can be expa. nded about those of a beam with zero taper in terms of the taper parameters. Rao and Carnegie [5] used the finite difference method to obtain the frequencies and mode shapes of tapered blading. MaNe and Rogers [6, 7] solved the differential equation of vibration of tapered beams with different boundary conditions using Bessel functions and tabulated the results of the first five vibrational frequencies for different breadth and depth taper ratios. In analyzing pretwisted beams, different approaches have been used by various investiga- tors. Mendelson and Gendler [8] used station functions while Rosard [9] applied the Mykle- stad method. The Rayleigh-Ritz method was used by Di Prima and Handelman [10], Carnegie [11] and Dawson [12]. Rao [13] analyzed pretwisted beams using the Galerkin method. Carnegie and Thomas [14] used a finite difference procedure for the analysis of pretwisted beams. The finite element technique has also been applied by many investigators, mostly for the vibration analysis of beams of uniform cross-section. All these investigations differ one from the other in the nodal degrees of freedom taken for deriving the elemental stiffness and mass matrices. McCalley [15] derived consistent mass and stiffness matrices by selecting the 187

Finite element eigenvalue analysis of tapered and twisted Timoshenko beams

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Journal of Sound and Vibration (I 978) 56(2), 187-200

FINITE ELEMENT EIGENVALUE ANALYSIS OF TAPERED

AND TWISTED TIMOSHENKO BEAMS

R. S. GUPTA

Department of Mechanical Engineering, Punjab Engineering College, Chandigarh-11, India

AND

S. S. RAO

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur-208016, India

(Received 13 January 1977, and in revised form 14 September 1977)

The stiffness and mass matrices of a twisted beam element with linearly varying breadth and depth are derived. The angle of twist is assumed to vary linearly along the length of the beam. The effects of shear deformation and (otary inertia are considered in deriving the elemental matrices. The first four natural frequencies and mode shapes are calculated for cantilever beams of various depth and breadth taper ratios at different angles of twist. The results are compared with those available in the literature.

1. INTRODUCTION

The analysis of tapered and twisted beams has wide application in many industrial problems. The vibration and deflection analysis of compressor blades, turbine blades, aircraft pro- peller 131ades, helicopter rotor blades, gear teeth, springs of electromechanical devices, electrical contact switches, etc., all can be made by using such beam elements.

Tapered beams have been analyzed by many investigators using different techniques. Rao [1] used the Galerkin method to calculate the fundamental natural frequencies of beams tapered in depth. Housner and Keightley [2] applied the Myklestad [3] procedure to determine the first three modes of a tapered beam. Martin [4] obtained the frequencies of a tapered beam using the assumption that the eigenvalues and eigenvectors of a tapered beam can be expa. nded about those of a beam with zero taper in terms of the taper parameters. Rao and Carnegie [5] used the finite difference method to obtain the frequencies and mode shapes of tapered blading. MaNe and Rogers [6, 7] solved the differential equation of vibration of tapered beams with different boundary conditions using Bessel functions and tabulated the results of the first five vibrational frequencies for different breadth and depth taper ratios.

In analyzing pretwisted beams, different approaches have been used by various investiga- tors. Mendelson and Gendler [8] used station functions while Rosard [9] applied the Mykle- stad method. The Rayleigh-Ritz method was used by Di Prima and Handelman [10], Carnegie [11] and Dawson [12]. Rao [13] analyzed pretwisted beams using the Galerkin method. Carnegie and Thomas [14] used a finite difference procedure for the analysis of pretwisted beams.

The finite element technique has also been applied by many investigators, mostly for the vibration analysis of beams of uniform cross-section. All these investigations differ one from the other in the nodal degrees of freedom taken for deriving the elemental stiffness and mass matrices. McCalley [15] derived consistent mass and stiffness matrices by selecting the

187

188 R. S. G U P T A AND S. S. RAO

total deflection and the total slope as nodal co-ordinates. Archer [16] analyzed various beams with specific boundary conditions. Kapur [17] took bending deflection, shear deflection, bending slope and shear slope as nodal degrees of freedom and derived the elemental matrices of beams with linearly varying inertia. Carnegie et aL [18] analyzed uniform beams by consi- dering few internal nodes in it. Nickel and Secor [19] used total deflection, total slope and bending slope of the two nodes and the bending slope at the mid-point of the beam as the degrees of freedom to derive the elemental stiffness and the mass matrices of order seven. Thomas and Abbas [20] analyzed uniform Timoshenko beams by taking total deflection, total slope, bending slope and the derivative of the bending slope as nodal degrees of freedom.

In this work the finite element method is applied for finding the frequencies of natural vibration of doubly tapered and twisted beams. The stiffness and mass matrices of the beam element are developed by taking bending deflection, bending slope, shear deflection and shear slope in two planes as nodal degrees of freedom. The effects of shear deformation and rotary inertia, which are of significant importance at higher modes of vibration, are consid- ered in ~he derivation. The elemental matrices of a doubly tapered beam without pretwist and a pretwisted beam without taper can be derived as special cases of the present matrices. The stiffness and mass matrices of a tapered beam and that of a uniform beam without shear deformation are also special cases of the general matrices. The first four natural fre- quencies of vibration have been calculated for untwisted and a pretwisted doubly tapered cantilever beams by using the finite element thus developed. The effects of variation of depth and breadth taper ratios of the beam have also been studied. The results compare well with those reported in the literature.

2. ELEMENT STIFFNESS AND MASS MATRICES 2.1. DISPLACEMENT MODEL

Figure l(a) shows a doubly tapered, twisted beam element of length /with the nodes as 1 and 2. The breadth, depth and the twist of the element are assumed to be linearly varying along its length. The breadth and depth at the two nodal points are shown as bt, lh and b2, h2, respectively. The pretwist angles at the two nodes are denoted by 01 and 02, respectively (a list of notation is given in Appendix B). Figure l(b) shows the nodal degrees of freedom of the element, with bending deflection, bending slope, shear deflection and shear slope in the two planes taken as the nodal degrees of freedom.

The total deflections of the element in the y and x directions at a distance z from node 1, namely, w(z) and r(z), are taken as

w ( z ) = w~(z) + we(z) , v (z ) - - v~(z) + v~(z ) , ( l )

where wb(z) and vb(z) are the deflections due to bending in the y z and x z planes, respectively, and w~(z) and v~(z) are the deflections due to shear in the corresponding planes.

The displacement models for wb(z) , w~(z), vb(z) and v~(z) are assumed to be polynomials of third degree. They are similar in nature except for the nodal constants. These expressions are given by

wb(z ) = ( u l / I 3) (2z a - 3 1 z 2 + / 3) + (u2/ l a) (31z 2 - 2z 3) -

- ( u 3 / p ) ( z 3 _ 21z 2 + t 2 z ) _ (u4/ l 2) ( z 3 _ l z2) ,

Ws(z ) = (us/13)1"2z 3 _ 3 l z 2 - 13) + (u6/l 3) (31 z 2 - 2 z 3) -

- (uT/ t ~) ( z 3 - 21z" + 12z ) - ( u s / t ~) ( ~ - t z2 ) ,

v~(z) = ( , 19 / l 3) ( 2 z 3 - 31z 2 - 13 ) + (Ulo/.13i ( 3 / z 2 _ 2 z 3) _

- ( u l J l Z ) ( z 3 - 2 z 2 + 12z ) - (u12/12)(z 3 - l z2 ) ,

v , ( z ) = (u13/ 3) (2z3 - 3 z 2 - 13) + ( u , , / 1 3 ) ( 3 1 z 2 - 2z 3) -

- ( u x s / l 2) (z 3 - 2 l z 2 + l ' z ) - ( u z , / l 2) ( z 3 - 1z2), (2)

TAPERED AND TWISTED TIMOSHENKO BEAMS 189

Ul U5 uZ U6

L, /" u / l ~ u B

I uII / u /

/ : Y'~ IY / /

u~ UI3 ~ 1o x r u14

u \ y '

(c)

Figure 1. (a) An element of tapered and twisted beam; (b) degrees of freedom of art element; (c) angle of twist 0.

where ul, u2, ua and tt4 represent the bending degrees of freedom and us, u6, u7 and Us are the shear degrees of freedom in the y z plane, it9, trio , till and u~2 represent the bending degrees of freedom and u~3, ttz4, it15 and u16 the shear degrees of freedom in the x z plane.

2.2. ELEMENT STIFFNESS MATRIX

The total strain energy U of a beam of length 1, due to bending and shear deformation, is given by

U = o LI. 2 k OzZ] + E l , , , Oz z Oz 2 ~- 2 kOz z / j + - - 2 - t k W ] + k ~ z ] j l d z . (3)

As the cross-section of the element changes with z and as the element is twisted, the cross- sectional area A, and the moments of inertia I,`,`, I . and I,`y will be functions o f z:

A(z ) = b ( z )h ( z ) = {bz + (b, - b , ) z / l } {hx + (h2 - h z ) z / l } = ( l /12) (c , z* + cz l z + c3 12), (4)

where

cl = (b2 - b,) (h2 - hi), c2 = b~(h2 - hx) + ht(b2 - b~),

I ~ ( z ) = I,`, ,`, cos z 0 + Ir y. sin z 0,

I . (z ) = Iy.y, cos 2 0 + 1,`,,`, sin 2 0,

I,`r(z) = (Ix,,`, - Iy, y,) �89 sin 20,

c3=blhl, (5)

(6)

190 R.S. GUPTA AND S. S. RAO

where x ' x ' and y'y ' are the axes inclined at an angle O, the angle o f twist, at any point in the element, to the original axes x x and yy as shown in Figure l(c). The value of /=, , , = 0 and the values ofl~,~, and I,,y, can be computed as

b(~)h'(~) I Ix,~,(z)-- l'-----~- = 12Y [a ' z4+a2 I z3+a312z2+a413z+as] ' (7)

where

where

a, = (b2 - b,) (h2 - h,) 3, az = b,(h2 - h,) 3 + 3(b2 - b,) (h2 - h,) 2 h,,

a3 = 3{b,h~(h2 - h,) 2 + (62 - b,) (h2 - h~)h~},

a, = 3b, h~(h2 - h,) + (b2 -- 6,) h~, as = bx h~,

~,. ,.(z) = h(z). b3(z) = ~ [a, z" + a2 t : + a~ t" z 2 + d, t 3 z + a, t'], 12 1214

(8)

(9)

the element stiffness matrix can be expressed as

[AK] [0] [DK] [0]

[o] [CK] [0] [01 1, [K] = [DK] [0] [BK] [0] / 1 6 x 1 6

[o1 [o1 [o1 [CKl ]

and

i la 2 wb\ [a" vb \

where [AK], [BK], [CK] and [DK] are symmetric matrices o f order 4 and [0] is a null matrix o f order 4. The elements o f matrices [AK], [BK], [CK] and [DK] are formulated in Appendix A.

0 5 )

(16)

d, = (h2 - h,) (62 - b,) 3, d2 = h,(b2 - b,) 3 + 3(h2 - h,) (62 - b,) 2 b,,

d3 = 3{h, b,(62 - 6,) 2 + (h2 - h,) (b2 - 6,) 6~},

d4= 3h, b ~ ( b 2 - b , ) + ( h 2 - h , ) b ~ , d s - -h ,b] . (113)

By substituting the expressions for wb, w:, vb, vs, A, Ixx, Ixy and Iyy f rom equations (2), (4) and (6) in equat ion (3), the strain energy U can be expressed as

v = �89 (l l)

where u is the vector o f nodal displacements u,, u2 . . . . . u,6, and [K] is the elemental stiffness matr ix o f order 16. In terms o f the integrals defined as

f [a 2 w~\' EI,,,,~--~z~J dz= [u~u2u3u4lr[AK][u~u2u3u4], (12)

z /02. \2

i . l a w , \ 2 ~AG \ //-~-z / dz = [u, u6 u7 us] r [CK] [u, u6 u7 us], (14) 0

TAPERED AND TWISTED TIMOSHENKO BEAMS 191

2 .3 . ELEMENT MASS MATRIX

The kinetic energy of the element T, including the effects of shear deformation and rotary inertia, is given by

o;L w + p /a'w~\/azv~'~ p[= [a~w,\q

+gi,,,ta---~t)t-~.~]+.~glO--~) ldz . (17)

By defining

and

! 2

J -~- \ ~ - / dz = [a, a~ a3 a,]" [aM] [a, a, a3 a,], 0

l 2 ( 3 T \ 0--ff~/ dz = [a, uz a, ~,]r [BM] [a, uz u3 ti,], 0

!

f T \a-~-~/ dz = [a9 a,o a,, a,=]" [CM] [ag.,o.,, a,.]. o

(18)

(19)

(20)

It is to be noted here that all the forced boundary conditions could be satisfied by the present model. Among the natural boundary conditions, if the condition of zero bending moment is to be enforced at a free end, the element due to Thomas and Abbas [20] is expected to be better than the present one.

free end: OwdOz=O and Ov,/Oz=O; (24)

clamped end: w, = 0, wb = 0, v, = 0, vb = 0, Owb/Oz = 0 and OvdOz = 0; (25)

hinged end: w, = 0, wb = 0, v, = 0 and vb = 0. (26)

conditions:

1

Jo g \Oz Ot] ~Oz Ot] dz = [a, az z~3 9,]r [DM] [99 9,o u,, ~,=1, (21)

where ~ denotes the time derivative of the nodal displacement tq, i = 1, 2 . . . . . 16, the kinetic energy of the element can be expressed as

T = �89 fi, (22) where [M] is the mass matrix given by

[ [AM] + [aM] [AM] [DM] [01 ] / /

= [ [AM] [AM] [AM] [01 1 [M] (23) / [DM] tAM] tAM] + [CM] tAM]/' l~6x16

/ /

L [0] [0] [AM] [AMlJ

and [AM], [BM], [CM] and [DM] are symmetric matrices of order 4 whose elements are defined in Appendix A.

2.4. BOUNDARY CONDmONS

The following boundary conditions are to be applied depending on the type of end

192 R.S. GLIPTA AND S. S. RAO

3. NUMERICAL RESULTS

The element stiffness and mass matrices developed are used for the dynamic analysis of cantilever beams. By using the standard procedures of structural analysis, the eigenvalue problem can be stated as

(iX] -- o,2[M]) U = 0, (27)

where [K] and [M] denote the stiffness and mass matrices of the structure, respectively, U indicates nodal displacement vector of the structure, and w is the natural frequency of vibration.

A study of the convergence properties of the element is made by taking the special case of a uniform beam with a length of 0.2540 m, breadth of 0.0762 m, depth of 0.0704 m, E = 2.07 • 10 ~I N /m z, G = 3E/8, mass density of 800 kg/m 3,/a = 2/3 and 0 = 0 ~ For this beam, the first, second, third and fourth natural frequencies obtained by the present method (with 4 elements) have been found to have 0"00y o, 0.07 ~o, 0.30 ~ and 0.60 ~ errors, respectively. The first four natural frequencies obtained by using 8 elements are 845.8, 3989.5, 8836.8 and 13827.1 Hz while the exact values are 845.8, 3988.9, 8834.2 and 13818.1 Hz, respectively [20]. The convergence of the natural frequencies of a pretwisted doubly tapered cantilever beam has also been studied and the results are shown in Table 1. In this case the natural frequencies given by the method of reference [21] have been found to be slightly higher than those predicted by the present method. It can also be seen that reasonably accurate results can be obtained even by using four finite elements.

TA~Lz 1

Natural frequencies of a tapered and twisted Tirnoshenko beam (Hz)

Number of elements First mode Second mode Third mode Fourth mode

1 304-8 1187.0 2259.3 4519-2 2 298.7 1146.8 1685-3 4046.5 3 298-1 1139.2 1652.2 3647.4 4 297.9 1137.9 1647.3 3593.5 5 297.8 1137.5 1646.0 3585.6 6 297.8 1137.4 1645.3 3578.8 7 297.8 1137.3 1645.1 3578.5 8 297-8 1137-3 1645-0 3578.3

Accordingto the method of re~rence [21] 299.1 1142-8 1653"3 3595.7

Data: length of beam = 0.1524 m, breadth at root = 0-0254 m, depth taper ratio = 2.29, breadth taper ratio = 2"56, twist = 45 ~ shear coefficient = 0.833, mass density = 800 kg]m 3, E= 2.07 x 10 tl N]nl 2, G = (3/8)E.

Table 2 shows the frequency ratios of an untwisted tapered beam for various combinations of depth and breadth taper ratios. Six finite elements are used to model the beam. It is observed that for constant depth taper ratio the frequency ratio of all the four modes increases with breadth taper ratio while for constant breadth taper ratio the frequency ratio decreases for the first mode and increases for the second, third and fourth modes with an increase in the depth taper ratios. The shear deformation effects reduce the frequency of modal vibration. The present results can be seen to compare well with those reported by Mabie and Rogers [6] which are also indicated in Table 2. Figure 2 shows comparison of the results given by the finite element method with those reported by Rosard [9] for a twisted beam of 0.0254 m x 0.00635 m cross-section and 0.2794 m length. I t can be seen that the two sets of results are quite comparable.

77

II 0

II

g II

I1

,+--

' 0 , .~

Z

0

0

lJ

II

"0

00 0 II

E

~J

",d"

o 0 II

o

~J

cJ ~J

,o

~o ~

~j~3

.~0~ 0 ~

0 . . . . . .

~j 0 0 0

194 R. S. GUPTA AND S. S. RAO

9

8 -

7

6

.o

g 4 ~

3

2

I

1 I

Third mode

Second mode

First mode

I I I0 ZO

Angle of twist (degrees)

30

Figure 2. Comparison of results for an uniform twisted beam. - - - - , Values by Rosard method; - - values by present method.

Figures 3 and 4 show the variation of modal frequencies with breadth taper ratio for beams having 0 ~ 30 ~ 60 ~ and 90 ~ twist with constant depth taper ratio while Figures 5 and 6 show similar variations for beams with constant breadth taper ratio and varying depth taper ratio. Here the length of the beam is taken as 0.254 m and the root cross-section as 0.076 x 0.038 times the length of the beam. Again the effects of breadth and depth tapers are seen to

I i I

4

r - Second mode

O~ o

6O*

F First mode / ] / - 9 o ~

' ///// / /s . /llO ////ll ~ - . ~ . ~ f i l l / 1 ~ . ~ , - ~ - - = - -

I I I 2 3 4

Breadth toper rQliO

Figure 3. Effects of shear deformation and breadth taper ratio on the first and second natural frequencies of a twisted beam. e, Method of reference [22]; , without shear deformation; . . . . , Timoshenko beam; depth taper ratio = = 3.

T A P E R E D A N D T W I S T E D T I M O S H E N K O BEAMS

14 I I I

13 \ \ \ \ ~.\

12 \ \ \

/ f / f - Fourth mode 11 - \ \ \ ~'',- -

0 I0 ~ . ~

9 - ~"

t~ 8 /f/f-Third mode

~ +

6 - 30"

4 I i I I 2 3 4 5

Breadth toper ratio, fl

195

Figure 4. Effects of shear deformation and breadth taper ratio on the third and fourth natural frequencies of a twisted beam. , Without shear deformation; . . . . . , Timoshenko beam; depth taper ratio cr = 3.

be p ronounced at higher modes of vibrat ion. Here also the effect of shear deformat ion is seen to reduce the modal frequencies at higher rates in higher modes of vibrat ion in all the

cases. The results found by the method of Carnegie and Thomas [22] for the first two natura l

frequencies are also indicated in Figures 3 and 5. It can be seen that the present results compare excellently with those of Carnegie and Thomas.

.9

u~

I I I

~ 90 ~ Firsl mode --~ I~, 60 ~

I I I 2 3 4

Depth taper ratio, a

Figure 5. Effects of shear deformation and depth taper ratio on the first and second natural frequencies of a twisted beam. , Without shear deformation; . . . . , Timoshenko beam; breadth taper ratio fl= 3. o, Method of reference [22].

196

a twisted beam.

13

12

II

I0

._o 9

7

6

5

4

R. S. G U P T A AND S. S. RAO

I I I

FcxJrth mode~ ~ ~)o

Thi rd mode --~ o

i i I 2 3 4

Depth taper ratio

Figure 6. Effects of shear deformation and depth taper ratio on the third and fourth natural frequencies of , Without shear deformation; . . . . . , Timoshenko beam; breadth taper ratio/Y = 3.

4. CONCLUSION

The finite element procedure developed for the eigenvalue analysis of doubly tapered and twisted Timoshenko beams has been found to give reasonably accurate results even with four finite elements. The effects of breadth and depth taper ratios, twist angle and shear deforma- tion on the natural frequencies of vibration of cantilever beams have been investigated. The present results are found to compare very well with those reported in the literature. The element developed is expected to be useful for the dynamic analysis of blades of roto-dynamic machines.

REFERENCES

1. J. S. RAo 1965 Aeronautical Quarterly 16, 139-144. The fundamental flexural vibration of cantilever beam of rectangular cross-section with uniform taper.

2. G. W. HOUSNER and W. O. KEIG}{T[.EY 1962 Proceedings of the American Society of Civil Enghz- eers 88, 95-123. Vibrations of linearly tapered beam.

3. N. O. MYKLESrAD 1944 Journal of Aerospace Science 2, 153-162. A new method for calculating natural modes of uncoupled bending vibrations of aeroplane wings and other types of beams.

4. A. I. MARTIN 1956 Aeronautical Quarterly 7,109-124. Some integrals relating to the vibration of a cantilever beams and approximations for the effect of taper on overtone frequencies.

5. J. S. RAO and W. CARNEGIE 1971 Bulletin of Mechanical Engineerhtg Education 10, 239-245. Determination of the frequencies of lateral vibration of tapered cantilever beams by the use of Ritz-Galerkin process.

6. H.H. MAB1E, and C. B. ROGERS 1972 Journal of the Acoustical Society of America 51, 1771-1774. Transverse vibrations of double-tapered cantilever beams.

7. H. H. MABm and C. B. ROGERS 1974 Journal of the Acoustical Society of America 55, 986--988. Vibration of doubly tapered cantilever beam with end mass and end support.

8. m. MENDELSON and S. GENDLER 1949 NACA TN-2185. Analytical determination of coupled bending torsion vibrations of cantilever beams by means of station functions.

9. D. D. ROSARD 1953 Jottrttal of Applied Mechanics 20, 241-244. Natural frequencies of twisted cantilevers.

10. R. C. DI PRIMA and G. H. HANDELMAN 1954 Quarterly on Applied Mathematics 12, 241-259. Vibration of twisted beams.

TAPERED AND TWISTED TIMOSHENKO BEAMS 197

11. W. CARNEGIE 1959 Proceedings of the Institute ofitlechanical Engineers 173, 343-346. Vibration of pretwisted cantilever blading.

12. B. DAWSON 1968 Journal of Mechanical Engineerhtg Science 10, 381-388. Coupled bending vibrations of pretwisted cantilever blading treated by Rayleigh-Ritz method.

13. J. S. RAG 1971 Journal of the Aeronautical Society of lndia 23, 62-64. Flexural vibration of pretwisted beams of rectangulc.r cross-section.

14. W. CARNEGIE and 3". THOMAS 1972 Journal of Engineerhtg for Industry, Transactions of the. American Society of Mechanical Enghwers 94, 255-266. The coupled bending-bending vibration of pretwisted tapered blading.

15. R. McCALLEY 1963 General Electric Company, Schenectady, New York, Report No. DIG/SA, 63-73. Rotary inertia correction for mass matrices.

16. J. S. ARCHER 1965 American Institute of Aeronautics and Astronautics Journal 3, 1910-1918. Consistent matrix formulations for structural analysis using finite element techniques.

17. K. K. KAPUR 1966 Journal of the Acoustical Society of America 40, 1058-1063. Vibrations of a Timoshenko beam, using finite element approach.

18. W. CARNEGIE, J. THOMAS and E. DOCUMAKI 1969 Aeronautical Quarterly 20, 321-332. An improved method of matrix displacement analysis in vibration problems.

19. R. NICKEL and G. SECOR 1972 bzternational Journal of Numerical Methods in Enghwering 5, 243-253. Convergence of consistently derived Timoshenko beam finite elements.

20. J. THOMAS and B. A. H. ABBAS 1975 Journal of Soundand Vibration 41,291-299. Finite element model for dynamic analysis of Timoshenko beam.

21. 3. S. RAG 1972 Journal of Engineerhtg for bMustry, Transactions of the American Society of Mechanical Engineers 94, 343-346. Flexural vibration of pretwisted tapered cantilever blades.

22. W. CARNrGm and J. THOMAS 1972JournalofEngineeringforlndustry, Transactions of the Ameri- can Society of Mechanical Engineers 94, 367-378. The effects of shear deformation and rotary inertia on the lateral frequencies of cantilever beams in bending.

APPENDIX A: EXPRESSIONS FOR [AK], [BK] . . . . , [DM]

The following notation is used for convenience:

1

w,=Jz'-'dz, i = 1,2 . . . . . n, (AI) 0

L l = I t - l , i = 1,2 . . . . . n, (A2)

[ z ] Vl = f z ' - ' cos 2 (02 - 0,) + 01 dz, i = 1,2 . . . . . n, (13)

0

, [ z ] St = f z l-~ sin 2 (02 - 01) + 01 dz, i = 1,2 . . . . . n, (14)

0

where O~ and 02 denote the values of pretwist a t nodes I and 2, respectively, of the element. As wb, w~, Vb and v~ are all the same in nature except for their positions in the stiffness and

mass matrices, one can use )~ to denote any one of the quantities wb, ws, vb or vs and in a similar manner the set (111,112,113, a4) can be used to represent any one of the sets (u~, u2, u3, u4), (us, u6, u7, us), (ug, U,o, ul,, u12) or (u13, u14, u,s, U16 ). Thus

1] 13 _113 -3 or.2 12Z) q_~3(3122_223)_~__~4(23 - if(z)=~-~(2z 2 - 3 1 z 2+ ) ~ ( ~ - , . , ~ + 1 12" IZ2), (AS)

d~ 111 ~ (3Z 2 - - 41z " - -~z = -f~(6z2-61z) - + 1 2 ) + ~ ( 6 1 z - 6 z 2 ) - ~ ( 3 z 2 - 2 1 z ) , (A6)

d 2 if, l] 1 . a3 t]2 a 4 dz 2 = I-S (12z 61) - . ~ ( 6 z - 41) + i f (61 - 12z) - 7~(6z - 21). ( t 7 )

198 R. S. GUPTA AND S. S. RAO

By lett ing P~,t.k(i = 1 . . . . . 4 ; j = i , . . . , 4 ; k = t . . . . . 7) denote the coefficient ofzk-Xl 7-k for the a~zT~ term in the expression o f ~2, Q~.~,k(l = 1 . . . . . 4; j = i . . . . . 4; k = 1 . . . . . 5) the coefficient o f zk-~l 5-k for the f i ~ term in the expression o f ( d ~ ] d z ) z, R~.j,k(i = 1 . . . . . 4 ; j = i, . . . . 4; k = 1 . . . . . 3) the coefficient ofzk-~l 3-k for the a~aj te rm in the expression o f (d 2 ~'/dz2) 2, Hi. j( i = I . . . . . 4 ; j = i . . . . . 4) the index coefficient o f l to account for the differ- ence in index o f I due to mul t ip l ica t ion o f ro ta t iona l degrees o f f reedom ~1 and a2 and the d isplacement degrees o f f reedom a3 and t74, the values ofP~.j.k, Q~.j.k, R~.j,k and H~,~ can be ob ta ined as shown in Tables A I and A2.

TABLE A1

Vahtes o f Hi.j , Ri.j.k, Ql.j.~

R~.~., for k = Qt. j . , for k = )k A

r "~ r �9

i j H~.j 1 2 3 1 2 3 4 5

1 1 0 144"0 -144"0 36"0 36.0 -72-0 36"0 6-0 0.0 1 2 0 -144"0 144"0 -36 .0 -36"0 72"0 -36"0 0-0 0'0 1 3 1 -72"0 84"0 -24 .0 -18 .0 42"0 -30"0 6'0 0'0 1 4 1 -72"0 60"0 -12"0 -18"0 30'0 -12"0 0"0 0.0 2 2 0 144-0 -144"0 36"0 36-0 -72"0 36"0 0-0 0-0 2 3 1 72-0 -84"0 24.0 18:0 -42"0 30"0 -6 -0 0'0 2 4 1 72-0 -60 '0 12"0 18"0 - 3 0 ' 0 12"0 0-0 0.0 3 3 2 36-0 -48"0 16"0 9"0 -24"0 22.0 -8-0 1"0 3 4 2 36"0 -36-0 8.0 9"0 -18"0 11.0 -2"0 0"0 4 4 2 36-0 -24"0 4-0 9"0 - I 2 - 0 4-0 0-0 0"0

TABLE A2

Vahles o f Pt, j.

PI.j.~ for k = r

i ] I 2 3 4 5 6 7

I 1 4.0 -12 .0 9.0 4.0 -6 .0 0.0 1.0 1 2 - 4 . 0 12.0 - 9 . 0 - 2 , 0 3.0 0.0 0-0 1 3 - 2 . 0 7.0 - 8 . 0 2.0 2.0 -1 -0 0-0 1 4 - 2 . 0 5.0 - 3 , 0 - 1 . 0 1.0 0.0 0-0 2 2 4.0 -12 .0 9.0 0-0 0.0 0.0 0.0 2 3 2.0 -7 .0 8.0 -3 .0 0.0 0.0 0.0 2 4 2-0 -5 .0 3.0 0.0 0.0 0.0 0-0 3 3 1-0 - 4 . 0 6-0 -4 -0 1-0 0-0 0-0 3 4 1.0 -3 .0 3.0 -1 .0 0-0 0.0 0.0 4 4 1.0 -2 .0 1-0 0.0 0.0 0.0 0.0

EVALUATION OF [BK]

As the p rocedure for the der iva t ion o f [AK], [BK] . . . . . [D~I] is the same for each, the express ion for [BK] is der ived here as an i l lustrat ion. One has

(A8)

TAPERED AND TWISTED TIMOSHENKO BEAMS 199

where ~ = vb and

tt2 = Ulo .

t23 ) u H 94 ku12

Putting the value oflyr and ff in equation (A8) gives

i /a2rv\' El ,%. + (I,.,. {(0, 0 z+ x i..>co,, , , , 0,)]

x g(12z - 61) - ~(6z - 40 +~(6! - 12z) - ~7(6z - 20

with

dz, (A9)

BKx. a = coefficient of a~ ax = coefficient of u9 u9

E

1211~

+ <(dl z 4 + d2 l z 3 + d3 1 z z 2 + d4 l 3 z + ds) - (al z 4 + a2 l z 3 + a3 l 2 z 2 +

, , , + , , , , , + ,

E s - 121--i-~L(n,.,+x ) ~. {a,[R,.,. a Us-, + L , + I R L , . 2 UT-, +L,+2Ra., .3 V6-,] +

l - I

+ (d, - a,) [L, Rl.l .1 Vs_, + L,+x R~ .x.2 VT_, + L,+2 R1.1.3 V6-,]}

g 5 3

-- 121--i-6L(n,., +t) ~ ~ [{aiLl+s-1 U g - t - j R l . 1,j} + (d, -- at){Li+j-t V g - I - j R t . 1.s}]. t=tj=a (AI0)

This relation can be generalized as

g 5 3

BKI . s = l - ~ 6 L ( n , . , + 1 ) ~. ~. [{a, Lu+.j-1) U(9-i-j) Rz,s.j} + 1=I J=l

+ (dr - al){Lo+s-a) V(9-1-.oRt.s , j}] , I = 1 . . . . . 4, J = 1, . . . . 4,

g 5

- 1 2 7 ~ E ~ [{aIL(I+J+HI.j )U9-1-JRI.J,J}+(dl-al)X t=t J - t

• { L u + j + U , , s ) V(9_l_j).Rl.s.j}], I = 1 . . . . . 4,

Similarly

E AK~.s = - -

12D ~

5 3

E [{dIL(i+J+HI"I)U(9-I-j)RI'j'J}+ 1=1 j = l

J = 1 , . . . , 4 . (All )

+ ( a , - d t ) { L u + s + n , . s ) V g _ , _ j R r . a . ~ } ] , I = 1 . . . . . 4, J = I . . . . . 4, (AI2)

2 0 0 R . S . G U P T A A N D S. S. R A O

3 5

CK, s = I t G ~ ~[c ,L( ,+j+n, .,, Uo- i - j ) Ot . j . j ], 1 = 1, 4, �9 18 ~ �9 . . . ,

l = l j = l

J = / , . . . , 4 , (AI3)

5 3

I=1 1=1

I = l . . . . . 4 , J = I , . . . . 4 ,

(AI4) 3 7

A M , . s = g ~ x ~ [ c , L , . s . m . j i U t u _ , _ s ) P , . , . , ] , I = I . . . . . 4, = J = l

5 5

BMt s = P ~ ~ [{d, Lc,+j+m ,) Uo,- , - j ) Qt.s.j} + �9 12gP ~

I - 1 / = t

+ (a, - dl){L(t+l.n,.:Vol_l_j) Qt.s.j}], I = 1 . . . . . 4,

$ 5 P

CMt s = �9 12gll----- ~ ~. ~ [{a,L,+j+,,,.,, Ucu_,_j, Q,.j.~} + / - 1 J = l

+ (di - al){L(i.~+n,.~) V(u-~-~) Q1.s.j}], I = 1 . . . . . 4,

5 5 _ P

1=1 1=1

J = I , . . . . 4, (Al5)

J = l . . . . . 4, (AI6)

J = I . . . . . 4, (A ! 7)

I = 1 . . . . . 4 , J = I , . . . . 4 .

(AI8)

APPENDIX B: NOMENCLATURE

A b E g G h

Ix,,, IT,),, Ix~, [K]

1 L

[M] t

U

U 1)

W

x,y Z

frequency ratio

O~

fl 0 P p

area of cross-section breadth of beam Young's modulus acceleration due to gravity shear modulus depth of beam moment of inertia of beam cross-section about xx, yy and xy axis, respectively element stiffness matrix length of an element length of total beam element mass matrix time parameter nodal degrees of freedom strain energy displacement in xz plane displacement in yz plane co-ordinate axes co-ordinate axis and length parameter ratio of modal frequency to frequency of fundamental mode of uniform beam with the same root cross-section and without shear deformation effects depth taper ratio, = hdha breadth taper ratio, = b~[b2 angle of twist mass density shear coefficient

Subscripts: b, bending; s, shear.