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Journal of Sound and Vibration (1996) 195(3), 507511

LETTERS TO THE EDITOR

A VARIATIONAL APPROACH TO THE VIBRATION OF TAPERED BEAMSWITH ELASTICALLY RESTRAINED ENDS

R. O. G B. V. A

PROMAS., Facultad de Ingenieria, Universidad Nacional de Salta, Buenos Aires 177.4400-Salta, Argentina

(Received 22 May 1995, and in final form 13 November 1995)

1.

The determination of natural frequencies in transverse vibrations of tapered beams withelastically restrained ends is a problem that has been extensively studied by severalinvestigators. To review the literature is not attempted here and only a few references willbe mentioned. Mabie and Rogers [14] studied several cases of tapered beams withdifferent end conditions. Laura and co-workers [58] treated various cases of non-uniformbeams with different conditions of end restraints. Grossi and Bhat [910] analyzed the caseof linearly tapered beams with ends elastically restrained against rotation. The problemof determination of frequencies for beams with both ends elastically restrained againstrotation and translation has been studied by Kameswara Rao and Mirza [11], but for thecase of uniform beams. The present work is concerned with the use of theRayleigh-Schmidt method in the determination of frequencies corresponding to the firsttwo modes of free vibration of a linearly tapered beam with both ends elastically restrainedagainst rotation and translation. It is shown that adopting, for the assumed mode shapes,

functions with several adjustable exponents leads to straightforward and simple algorithmwhich, in the case of the fundamental frequency coefficient, yields very accurate results.

An interesting feature of the present variational approach is that it increases the accuracyof numerical results through a refinement of the shape functions by optimizing theadjustable exponents rather than by increasing the terms of approximation. In thispaper, several examples are solved and the results obtained are compared withpreviously published results to demonstrate the accuracy and flexibility of the algorithmdeveloped. New results are also determined for tapered beams with generally restrainedends.

2. -

Consider a tapered beam of length lwhose ends are elastically restrained against rotationand translation. The width b(x) and depth h(x) vary linearly between x = 0 and x = l,where the point x = 0 corresponds to the left end of the beam. Let b(0)= b1, b(l) = b2,h(0)= h1 and h(l) = h2, then the variations of the b(x) and h(x) are given by

h(x) = h1(1 + c1x/l), c1 = h2/h1 1; 0Q xE l; (1a)

b(x) = b1(1 + c2x/l), c2 = b2/b1 1 (1b)507

0022460X/96/330507+ 05 $18.00/0 7 1996 Academic Press Limited

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The cross-sectional area and the moment of inertia are given by

A(x) = b(x)h(x) = A1(1 + c1x/l)(1+ c2x/l), A1 = b1h1; (2a)

I(x) = b(x)h3(x)/12 = I1(1 + c1x/l)3(1 + c2x/l), I1 = b1h31 /12; (2b)

A2 = A1(1 + c1)(1+ c2), I2 = I1(1 + c1)3

(1 + c2). (2c)The parameters b1, h1, A1 and I1 refer to the properties of the cross-section at the left, whileb2, h2, A2 and I2 refer to these properties at the right of the beam. In the case of normal

modes of vibrations, the maximum strain energy of the beam is given by

Ub,max =12 g

l

0

EI(x)u02(x) dx, (3)

where u(x) is the transverse deflection function. The maximum strain energy associatedwith the rotational restraints can be stated as

Ur,max =12 r1u'

2(0)+ 12 r2u'2(l). (4)

The maximum strain energy associated with the translational restraints can be written as

Ut,max =12 t1u

2(0)+ 12 t2u2(l). (5)

The total maximum strain energy of the system is given by

Umax = Ub,max + Ur,max + Ut,max . (6)

Finally, the maximum kinetic energy of the beam is given by

Tmax =12 rv

2 gl

0

A(x)u2(x) dx, (7)

where r is the mass density and v is the circular natural frequency. In the case of normalmodes of vibration the boundary conditions are [12]

r1u'(x) = EI(x)u0(x), x = 0; r2u'(x) = EI(x)u0(x), x = l; (8a, b)

t1u(x) = (EI(x)u0(x))', x = 0; t2u(x) = (EI(x)u0(x))', x = l; (8c, d)

It is convenient to change variables from x to x = x/lin equations (1)(8) in order to work

in the interval [0, 1]. Let us consider the following function for the assumed mode shapes:

u(x) = A1u1(x) + A2u2(x), (9a)

where

u1(x) = s4

i= 0

aixni, a4 = 1, n0 = 0, n1 = 1, n2 = 2; (9b)

u2(x) = s5

i= 0

bixmi, b5 = 1, m0 = 0, m1 = 1, m2 = 2. (9c)

The coefficients ai and bi in equations (9b, c) are determined from the boundary conditions

(8). The function u2(x) is subject to the additional requirement of having a nodal pointat x = 05. The exponents n3, n4, m3, m4 and m5 are the optimization parameters introducedin accordance with the Rayleigh-Schmidt method [910]. Substituting the deflection

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function, defined in equation (9) into equations (6) and (7) and minimizing the Rayleighquotient with respect to the coefficients A1 and A2 yields a frequency equation which canbe written as

l4(Ik1Ik2 I2k3) +l

2(2U3Ik3 U1Ik2 U2Ik1) + (U1U2 U23 ) = 0, (10)

where

l2 =rA1EI1

v2l4 (11)

is the non-dimensional frequency coefficient. From equation (10) one obtains two valuesofl2: l21 and l

22 . The non-dimensional frequency coefficient l

2, defined in equation (11) isa function of the adjustable exponents ni and mj introduced in equation (9). TheRayleigh-Schmidt method requires minimization of l2 with respect to the exponentialparameters. Trial variations of ni and mj in the neighborhoods of the values i and j aresufficient to determine approximate minima.

3.

Values of coefficients zl1 and zl2 of a uniform beam with ends elastically restrainedagainst rotation and translation are shown in Table 1. The values obtained with the presentmethod are compared with the exact values reported by Rao and Mirza [11]. Theagreement of values ofzl1 obtained with the Rayleigh-Schmidt method is excellent. Thismethod yields values which agree with the exact values within six significant figures (andin various cases seven). The values ofzl2 are not so accurate in all cases. The mentionedtable also depicts values ofzl1 and zl2 obtained with the Rayleigh-Ritz method withpolynomials expressions used as the assumed mode shape functions. In this case thecorresponding exponents are all fixed. It must be noted that the values ofzl1 obtainedwith this method are also very accurate. Table 2 contains values of the coefficient zl1obtained with fixed and adjustable exponents for a linearly tapered beam with different

T 1

Frequency coefficients zl1 and zl2 of a uniform beam with ends elastically restrainedagainst rotation and translation. (R1 = R2 = R, T1 = T2 = T). (I) Values obtained withRayleigh-Ritz method. (II) Values obtrained with Rayleigh-Schmidt method. (III) Values

from reference [11]

zl1 zl2ZXXXXXXXCXXXXXXXV ZXXXXXXXCXXXXXXXV

R I II III I II IIT=010

001 0668463 0668463 0668464 0956995 0956995 0956995010 0668473 0668473 0668473 1309212 1309212 13092111 0668539 0668539 0668539 2110992 2110992 2110991

10 0668655 0668655 0668655 2892034 2892006 2891977100 0668689 0668689 0668689 3114410 3114365 3114298T = 1 0

001 2032730 2032658 2032659 2768869 2768868 2768866010 2035451 2035385 2035385 2788462 2788462 27884581 2054061 2054026 2054026 2933289 2933280 2933271

10 2088264 2088257 2088258 3270997 3270939 3270873100 2098732 2098730 2098730 3403191 3403086 3403000

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510

T 2

Values ofzl1 of a tapered beam with different end conditions (b2 = b1). (I) Values obtainedwith Rayleigh-Ritz method. (II) Values obtained with Rayleigh-Schmidt method

h2/h1

End condition 02 025 033333 05 1R1 = T1 = a (I) 2096 2062 2018 1959 1875R2 = T2 = 0 (II) 2080 2047 2009 1955 1875R1 = T1 = a (I) 3257 3293 3361 3513 3929R2 = 0, T2 = a (II) 3138 3222 3326 3507 3926R1 = 0, T1 = a (I) 2296 2363 2470 2671 3142R2 = 0, T2 = a (II) 2237 2321 2452 2668 3141R1 = T1 = a (I) 3647 3700 3805 4048 4738R2 = T2 = a (II) 3518 3604 3752 4041 4730

classical end conditions. Finally, values of zl1 obtained with fixed andadjustable exponents, for a linearly tapered beam with ends elastically restrained against

rotation and translation, are included in Table 3.

4.

A simple, computationally efficient and accurate approximate method has been used todetermine numerical values of frequency coefficients of a linearly tapered beam with endselastically restrained against rotation and translation. Excellent agreement was obtainedbetween the present values ofzl1 and the comparison results. Table 1 shows that in thecase of uniform beams, the use of fixed exponents, that is the use of Rayleigh-Ritz method

with polynomials expressions as the assumed mode functions, also yields very accurateresults. Since both Rayleigh-Ritz and Rayleigh-Schmidt methods yield upper bounds for

T

3Values of zl1 of a tapered beam with ends elastically restrained against rotation andtranslation (R2 = 0, T1 = a, h2/h1 = 025, b2/b1 =05 and b2/b1 = 1). (I) Values obtained with

Rayleigh-Ritz method. (II) Values obtained with Rayleigh-Schmidt method

zl1ZXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXV

b2/b1 = 1 b2/b1 =05 b2/b1 = 1 b2/b1 =05ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV

R1 I II I II I II I II

T2 =000 T2 =01000 032193 032172 030080 03004901 090219 090200 100180 100150 090603 090574 100401 100361

10 195338 194044 215046 213050 195429 194110 215123 213095

100 205048 203481 225019 222614 205136 203544 225095 222659a 206219 204655 226179 223784 206306 204718 226254 223828T2 = 10 T2 = a

00 106415 101514 102179 095216 236301 232154 234082 22799201 119009 115137 121458 116844 239812 235653 238694 232640

10 204639 200323 223125 217527 310163 303750 321538 312459100 213995 209525 232944 227026 327145 319917 339476 329240

a 215052 210664 233995 228179 329341 322144 341670 331473

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the natural frequencies, it seems reasonable to expect that the values obtained with thelatter method are more accurate. Nevertheless, the observation that the use of fixedexponents leads to very accurate results, in the case of uniform beams with ends elasticallyrestrained against rotation and translation, is important since this procedure implies lesscomputational work. The new results for tapered beams with generally restrained ends

presented in Table 3 may conclude that both rotational and translational restraints havea significant effect on the first two frequencies and the translational restraints have greaterinfluence on these frequencies than the rotational restraint. It can be concluded that the

variational approach used constitutes a simple and unified analytical procedure which hasgreat flexibility and accuracy from a practical viewpoint when applied to the vibrationproblems considered.

This research has been partially supported by Consejo de Investigacion de laUniversidad Nacional de Salta. The authors are grateful to the reviewers for their valuablecriticism.

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2. H. H. M and C. B. R 1968 Journal of the Acoustical Society of America 44, 17391741.Transverse vibrations of tapered cantilever beams with end supports.

3. H. H. M and C. B. R 1972 Journal of the Acoustical Society of America 51, 17711774.Transverse vibrations of double-tapered cantilever beams.

4. H. H. M and C. B. R 1974 Journal of the Acoustical Society of America 55, 986991.Transverse vibrations of double-tapered cantilever beams with end support and with end mass.

5. P. A. A. L and R. H. G 1986 Journal of Sound and Vibration 108, 123131.Vibration of a elastically restrained cantilever beam of varying cross section with tip mass offinite length.

6. P. A. A. L, B. V. G, J. U and R. C 1988 Journal of Sound and Vibration120, 587596. Numerical experiments on free and forced vibrations of beams of non-uniformcross sections.

7. V. H. C and P. A. A. L 1985 Journal of Sound and Vibration 99, 144148. Vibrationand buckling of a non-uniform beam elastically restrained against rotation at one end and withconcentrated mass at the other.

8. S. A and P. A. A. L 1984 Applied Acoustics 17, 255260. Transverse vibrations ofbeams with variable moment of inertia and an intermediate support.

9. R. O. G and R. B. B 1991 Journal of Sound and Vibration 147, 174148. A note onvibrating tapered beams.

10. R. O. G, A. A and R. B. B 1993 Journal of Sound and Vibration 160, 175178.Vibration of tapered beams with one end spring hinged and the other end with tip mass.

11. C. K R and S. M 1989 Journal of Sound and Vibration 130, 453465. A noteon vibrations of generally restrained beams.

12. R. O. G 1988 International Journal of Mechanical Engineering Education 16, 5763. On thevariational derivation of boundary-value problems in the dynamics of beams and plates.