Journal of Sound and Vibration (1995) 185(4), 703716

VIBRATION OF NON-UNIFORM RODS ANDBEAMS

S A

Department of Technology, Southern Illinois University at Carbondale, Carbondale,Illinois 62901-6603, U.S.A.

(Received 21 March 1994, and in final form 23 August 1994)

We show that for some non-uniform rods and beams the equation of motion can betransformed into the equation of motion for a uniform rod or beam. Then, when the endsare completely fixed, the eigenvalues of the non-uniform continuum are the same as thoseof uniform rods or beams. For other end support conditions, exact solutions are obtained.An efficient procedure is used to analyze the free vibration of non-uniformbeamswith generalshape and arbitrary boundary conditions. Simple formulas are presented for predicting thefundamental natural frequency of non-uniform beams with various end support conditions.

7 1995 Academic Press Limited

1. INTRODUCTION

The vibration of beams and rods has been studied extensively, and is still receiving attentionin the literature. No attempt will be made here to present a bibliographical account ofprevious work in this area. Interest in the vibration of non-uniform one-dimensionalstructures was sparked by Eisenbergers study [1] of tapered rods showing that naturalfrequencies are only slightly affected by taper. In addition, Graf [2] mentioned that forrods with conical cross-sections the equation of motion can be put into the form of thewave equation by an appropriate change of variable. The study of conical rods is importantto the study of foundations [35] and the dynamics of composite structures subjected tohigh velocity impact [6]. In both of these cases, the dynamic response of a half-plane to asurface load can be accurately determined using a cone model. Therefore it was decided toinvestigate the topic further to determinewhen such a transformation is possible andwhethera similar result can be obtained for beams. Earlier investigations of the vibrations of taperedbeams include the study of Conway et al. [7] who obtained an exact solution to the equationof motion for a conical beam in terms of Bessel functions and presented determinantalequations for finding the natural frequencies of such beams for four sets of boundaryconditions. Mabie and Rogers [8] used the same approach to study clampedpinned beamswith constant width and linearly varying thickness, or constant thickness and linearlyvarying width. The natural frequencies of cantilever beams with linear tapered width andthickness were obtained numerically by Mabie and Rodgers [9, 10]. Goel [11] and Craverand Jampala [12] studied the vibration of linearly tapered beams constrained by a spring.Bailey [13] first used the RayleighRitz approach to study the vibration of generally taperedbeams.

In this study, exact solutions to a new class of tapered beams are presented. In order toverify these results, a general procedure for analyzing the free vibration of tapered beamsis developed. The RayleighRitz approach is used to formulate the problem, and thedisplacements are expanded in a series of polynomial approximation functions that can

703

0022460X/95/340703+14 $12.00/0 7 1995 Academic Press Limited

704

satisfy boundary conditions at x=0. The essential boundary conditions at the other end areenforced using the Lagrange multiplier method [14]. The cross-sectional area and themoment of inertia of the beam are assumed to be arbitrary polynomial functions ofposition. Results are presented for several particular cases, and show excellent agreementwith published results [1, 15, 16]. The present method is easy to implement and efficient.In addition, simple formulas to predict the fundamental frequencies are obtained for severalpractical cases using one-term RayleighRitz approximations. These formulas are shown tobe accurate, and should be of interest to designers.

2. NON-UNIFORM RODS

The axial motion of a rod with varying cross-section is governed by the differentialequation

1

1x $EA(x) 1u1x%=rA(x) 12u1t2 . (1)Graf [2] noted that when A(x)=x2, the differential equation can be put into the form

12(xu)1x2

=1c2

12(xu)1t2

, (2)

which means that after the change of variables v=xu, the motion of the rod is governed bythe classicalwave equation.Herewe seek all functionsA(x) forwhich the governing equation(1) can be put into the form

12[c(x)u]1x2

=1c2

12[c(x)u]1t2

, (3)

where c(x) is a function to be determined. Expanding both equations (1) and (3), we find

A'u'+au0=rE

A12u1t2

, (4)

c0u+2c'u'+cu0=c 12u

1t2, (5)

where a prime denotes differentiation with respect to x. Comparing equations (4) and (5),c0=0, which implies that

c=c1x+c2. (6)

Matching the next two terms in equations (4) and (5) gives

2c'A'

=c

A. (7)

From equations (6) and (7), we find that if the cross-section varies as

A=A0 01+a xL12

, (8)

705

where a is a non-dimensional parameter, then equation (1) can be transformed into equation(3) with

c(x)=1+axL

, c2=E/r. (9)

Equations (8, 9) define the type of non-uniform rods for which the equation of motion canbe transformed into the classical wave equation, as indicated in equation (3).

3. NON-UNIFORM BEAMS

The transverse motion of non-uniform beams is governed by

12

1x2 $EI(x) 12w1x2%+rA(x) 12w1t2 =0, (10)which we should like to transform to

EI014(cw)

1x4+rA0

12(cw)1t2

=0, (11)

where c is a suitable function of x. When this transformation is possible, exactsolutions for the vibration of non-uniform beams will be obtained, and we shall show thatclampedclamped beams have the same eigenvalues regardless of taper. Expandingequations (10) and (11) gives

E(I0w0+2I'w1+Iw2)+rA 12w

1t2=0, (12)

c2w+4c1w'+6c0w0+4cw0+cw2+12(cw)1t2

=0, (13)

where a prime denotes differentiation with respect to x. Comparing equations (12) and (13)indicates that, in order to match the left-hand sides, c2=c1=0, which implies that

c=a1x2+a2x+a3. (14)

Matching the terms on the left-hand side of equations (12) and (13) gives

6c0I0 =

4c'2I'

=c

I. (15)

To satisfy equation (15), we must have I=I0c2, the constants a1 and a2 must have the samesign, and a22=4a1a3. Therefore, we must take

I=I0(1+ax)4, A=A0(1+ax)4, c=(1+ax)2, (16)

where, as before, a is a positive constant and A0 and I0 are the cross-sectional area and themoment of inertia at x=0. When the properties of the beam vary according to equation (16),the equation of motion (10) can be put in the form of the equation of motion for a uniformbeam (11) by the change of variable v=cw.

706

4. RAYLEIGHRITZ APPROXIMATION

The RayleighRitz method is used to study the free vibration of non-uniform beamassuming an N-term approximation for the transverse displacements of the form

w=sN

i=1

aifi (x), (17)

where the approximation functions are taken to be polynomials satisfying the essentialboundary conditions at x=0. That is, fi=xi+d, where d=1, 0 and 1 for free, pinned andclamped support conditions at x=0 respectively. The natural frequencies and mode shapesare determined by solving the eigenvalue problem

([k]v2[m]){a}=0, (18)

where the coefficients of the stiffness and mass matrices are given by

kij=gL

0

EI(x)f0i f0j dx, mij=gL

0

rA(x)fifj dx. (19)

The approximation functions used cannot satisfy the essential boundary conditions at x=L.When necessary, these constraints are enforced using the Lagrange multiplier technique [14].With this approach, the stiffness and mass matrix coefficients can be easily obtained whenthe variation of cross-sectional area and moment of inertia can be described by polynomials.The moment of inertia and the cross-sectional area of the beam are assumed to vary as

I=I0 sn

k=0

ak 0xL1k

, A=A0 sn

k=0

bk 0xL1k

, (20)

where the ak and bk are constants and a0=b0=1. Substituting equations (17) and (20) intoequation (19), we obtain

kij=EI0ij(i1)( j1) sn

k=0

akLi+j3

i+j+k3, (21)

mij=rA0 sn

k=0

bkLi+j+1

i+j+k+1. (22)

With this formulation, the free vibration of a beam with any polynomial variation ofcross-sectional properties can be analyzed.

The RayleighRitz method is used to develop simple formulas for predicting thefundamental natural frequency for rods with cross-sections varying as

A=A0 01+a xL+b x2L21. (23)

707

For each case, assuming a polynomial approximation function that satisfies the essentialboundary conditions, the stiffness and mass coefficients are obtained as

k=gL

0

EA(f')2 dx, m=gL

0

rAf2 dx. (24)

For a pinnedpinned rod, taking f=x(x1), where x=x/L, we get

k=EA030L

(5a+4b+10), m= 1420rA0L(7a+4b+14). (25)

For a pinnedfree rod, taking f=x12x2, we get

k=EA060L

(5a+2b+20), m= 1840rA0L(77a+58b+112). (26)

Approximate solutions are also obtained using a one-term RayleighRitz approximationfor beams with an assumed polynomial variation of A and I as in equation (20). For thisapproximation, we take n=4 in order to cover the cases considered by Cortinez and Laura[8] and by Hodges et al. [9]. The stiffness and mass coefficients for four different sets ofboundary conditions are given in the appendix along with the corresponding approximationfunction f. Note that the fs are simple polynomial functions that satisfy all boundaryconditions.

5. AXIAL VIBRATION OF CONICAL RODS

First, consider rods with cross-sections varying according to equation (8). With the newvariable

v(x, t)=c(x)u(x, t), (27)

the equation of motion is the classical wave equation. Therefore, for free vibrations,

v=(A sin bix+B cos bix) sin vit. (28)

For a fixedfixed rod, the boundary conditions u(0, t)=u(L, t)=0 imply thatv(0, t)=u(L, t)=0, and therefore, the natural frequencies of the tapered rod are the sameas those of a uniform rod, and are given by vi=i(p/L)(E/r)1/2. In this case the mode shapesare given by

Ui (x)=sin bix1+ax

, (29)

which indicates that, even though the natural frequencies are independent of the parametera, the mode shapes are affected by taper.

For fixedfree rods, the boundary conditions u(0, t)=0 and 1u(L, t)/1x=0 lead to

v(0, t)=0,a

1+av(L, t)=

1v(L, t)1x

. (30)

Using equation (29), we obtain the transcendental equation

a

1+atan bL=bL, (31)

10

100

L

y

5

5

0

5

1 2 3 4 6 7 8

2

1

708

T 1

Non-dimensional natural frequencies bnL of fixedfree rods withA=A0(1+ax)2

Mode a=0 a=1 a=2

1 1570796 1165561 09674032 4712389 4604217 45674523 7853982 7789884 77683734 10995574 10949944 109346825 14137167 14101725 140898876 17278760 17249782 17240109

which can be solved for particular values of a, and then vi=biLc/L. Table 1 shows resultsfor uniform rods (x=0) and tapered rods with a=1, 2. For uniform rods, biL=12(2i1)p,and the results in Table 1 indicate that the lowest natural frequencies are affected most bytaper. For non-uniform rods, solutions to equation (31) can be interpreted as the values ofbiL for which the line y=(1+a)bL/a intersects the curve y=tan bL. Figure 1 illustrates whythe higher modes are less affected by taper. The curve y=tan bL has vertical asymptoteswhen bL is an odd multiple of 12p, so, for higher modes, the line y=(1+a)bL/a intersectsthe line y=tan bL for a value of bL very close to the asymptote, and the particular valueof a has little effect.

For freefree rods, the transcendental equation is

tan bL=a2bL

a2+(1+a)(bL)2, (32)

Figure 1. Graphical determination of natural frequencies of a rod with A=A0(1+ax)2. , y=tan bL; line2; y=2bL; line 2, y=bL/[1+2(bL2)].

709

T 2

Non-dimensional natural frequencies bnL of freefree rods withA=A0 (1+ax)2

Mode a=0 a=1 a=2

1 3141593 3286007 34743352 6283185 6360678 64800313 9424778 9477196 95613684 12566371 12605891 126703605 15707963 15739650 157921596 18849556 18875239 18918810

and the results for rods with a=0, 1 and 2 (Table 2) follows the same trend as for fixedfreerods: the lower modes are more affect by taper than the higher modes. Figure 1 shows howthe line y=bL/[1+2(bL)2] intersects y=tan bL to give the natural frequencies for a=1.Since the slope of that line is always very small, the natural frequencies are always very closeto np, and the effect is always small.

The boundary conditions for a bar that is fixed at x=0 and carries a concentrated massM at x=L are

u(0, t)=0, M12u1t2

(L, t)+EA(L)1u1x

(L, t)=0. (33)

Using equation (28), the transcendental equation for this problem is

(1+a)bL cos bL(a+Mb2L2) sin bL=0, (34)

where M=M/rAL is the ratio between the mass of the concentrated mass and the mass ofthe rod. Table 3 gives results for M=1 and several values of the taper coefficient a.

6. TRANSVERSE VIBRATION OF TAPERED BEAMS

Next, consider tapered beams with I and A given by equation (16), so that the equationof motion can be transformed into the equation of motion for a uniform beam by thechange of variable v=cw in equation (11). For free vibration, the general solution formode nis

v(x, t)=Vn (x) sin vnt, (35)

T 3

Non-dimensional frequencies bnL for rods fixed at x=0 with aconcentrated mass at x=L (M=1)

Mode a=0 a=1 a=2

1 0860334 0768217 07011062 3425619 3616202 37523583 6437298 6572276 66887134 9529334 9627477 97183385 12645287 12721368 127940096 15771284 15833121 15893097

710

with the mode shapes

Vn (x)=A sin bnx+B cos bnx+C ebnx+D ebnx, (36)

where the constants A,..., D are to be determined from the boundary conditions.For a clampedclamped beam, the boundary conditions

w(0, t)=1w(0, t)

1x=w(L, t)=

1w(L, t)1x

=0 (37)

yield

v(0, t)=1v(0, t)

1x=v(L, t)=

1v(L, t)1x

=0. (38)

Therefore, after transformation, tapered beams with properties varying as in equations (16)will have the same governing differential equation and the same boundary conditions as theuniform beam with the same A0 and I0. This implies that the natural frequencies of suchclampedclamped beams will always be the same regardless of the parameter a. This result,it appears, was verified with the general RayleighRitz approach developed earlier. Themode shapes of the tapered beam will then be given by Wn (x)=Vn (x)/(1+ax)2, in terms ofVn , the mode shape of the uniform beam (equation (36)). Therefore, while the eigenvaluesare independent of the parameter a, the mode shapes of the non-uniform beams are affectedby it.

For a clampedpinned beam, the boundary conditions

w(0, t)=1w(0, t)

1x=w(L, t)=

12w(L, t)1x2

=0 (39)

yield

v(0, t)=1v(0, t)

1x=v(L, t)=0,

12v(L, t)1x2

=4a

(1+a)L1v(L, t)

1x. (40)

Using equation (40), it can be shown that the boundary conditions are satisfied when bnLis selected to that

G G0 1 1 1G G1 0 1 1G G

sin bnL cos bnL ebnL ebnLG GG GbnL sin bnLC cos bnL bnL cos bnL+C sin bnL (bnLC) ebnL (bnL+C) ebnL

=0, (41)

with C=4a/(1+a). The natural frequencies, given by

vn=(bnL)20 EI0rA0L411/2

=Vn0 EI0rA0L411/2

, (42)

are obtained by first finding the values of bnL for which the determinant in equation (41)vanishes. The first six non-dimensional frequencies for a clampedpinned beam are givenin Table 4 for a=0, 1 and 2. As in the case of the tapered rod, the effect of taper is significantfor the lowest modes, but becomes small for nq3. Determinantal equations similar toequation (41) can be obtained for other types of boundary conditions.

Mabie and Rodgers [8] obtained exact solutions for the vibration of beams with constantheight and linear vibration in width, that is, for m=1 and a1=b1=a in equation (20).

711

T 4

Non-dimensional natural frequencies Vn of clampedpinnedbeams with A=A0(1+ax)4, I=I0 (1+ax)4 (numbers inparentheses are obtained using the one-term approximation)

Mode a=0 a=1 a=2

1 154182 123635 105984(154511) (137613) (136419)

2 499649 476265 4666783 104248 102025 1011744 178270 176105 1753045 272032 269904 2691366 385533 383423 382669

Recently, Hodges et al. [16] studied the vibration of such beam with clampedfree boundaryconditions using a finite elementtransfer matrix approach. Results obtained using thepresent RayleighRitz approach with 10 terms are very accurate when compared withthose in reference [16], as shown in Table 5. The one-term approximation also providesvery accurate results (12% error for a=05). For the pinnedpinned case, the results inthe appendix can be rewritten as

k=168EI035L3

(1+05a1+028571a2+017857a3+011905a4),

m=8866rA0L180 180

(1+05b1+028299b2+017449b3+011460b4). (43)

Similarly, for clampedclamped beams, the one-term approximation gives

k=84EI0105L3

(1+05a1+038095a2+032143a3+028571a4),

m=572rA0L360 360

(1+05b1+027273b2+015909b3+009790b4). (44)

For these two sets of boundary conditions, the one-term approximation suggests that thenatural frequencies should remain constant as a1 varies (a2=a3=a4=0). Figures 2(a) and (b)indicate that the one-term approximation is very accurate over a wide range of values

T 5

Non-dimensional natural frequencies Vn of cantilever beams with constant height andquadratically varying with (A/A0=I/I0=1+ax) (numbers in parentheses are obtained using

the one-term approximation)

a N Mode Present Hodges et al. [6] Thomson [17]

0 10 1 3516015269 35160(353009)

12 10 1 4315170298635 43151702986349(43673)

2 2351925663 631991974 12243963

16

0

1.0

1

14

12

2

0.5 0.0 0.5 1.0

2

1

4

6

8

10

(a)

3

4

40

5

35

10

2

1

15

20

25

30

(b)

3

4

8

0

1

62

1

2

4

(c)

3

4

25

5

10

2

115

20

(d)

3

4

1.0

0.5 0.0 0.5 1.0

712

Figure 2. First natural frequencies for non-uniform beams with (a) pinnedpinned, (b) clampedclamped, (c)clampedfree and (d) clampedpinned boundary conditions. 1, A/A0=I/I0=1+ax (exact); 2, A/A0=I/I0=1+ax(one-term); 3, A/A0=1+ax, I/I0=(1+ax)3 (exact); 4, A/A0=1+ax, I/I0=(1+ax)3 (one-term). Note the differentscales for V, in the four parts.

of a1 and that the first natural frequency does remain fairly constant over that range. Forcantilever beams and clampedpinned beams, the one-term approximation is also accurate,but the natural frequencies are strongly affected by taper (Figures 2(c) and (d)). Equations(43) and (44) also suggest that, if ai=bi for each value of i, the first natural frequencies oftapered beams should remain fairly constant. Figure 3 shows results for cantilever beamswith

a1=a2=a3=b1=b2=b3=a (45a)

30

0

1

2.5

25

10

5

0.5 1.0 1.5 2.0 3.0

21

20

15

4

3

5

713

Figure 3. First natural frequency of clampedclamped (1, cubic one-termapproximation; 2, cubic exact; 3, quarticexact) and pinnedpinned (4, cubic exact; 5, quartic exact) non-uniform beams.

T 6

Non-dimensional natural frequencies Vn of cantilever beams withconstant height and quadratically varying with (A/A0=I/

I0=1+x+x2)

n Mode Present Hodges et al. [16]

1 1 379375

2 1 2548012 3687240

3 1 2471312 2051069

4 1 247078872 19877273 6244077

5 1 24707868962 19845653 6077690

10 1 24707858401574 247078584015712 198446817253 5977406454

20 1 24707858401571 24707858401571(247801)

2 198446817250473 5977406374 119040848

714

T 7

Non-dimensional natural frequencies of beams with linearly varying height (A/A0=1+ax, I/I0=(1+ax)3) (numbers in parentheses are obtained using the one-term approximation)

BC a Mode Present Cortinez and Lawn [15] Thomson [17]

Cs 01 1 14848896 1485(149454)

2 476370373 99171635

0 1 15418206 1541 154182(154511)

2 49964862 4996453 10424770 1042477

01 1 159687099 1596(160004)

2 522372273 10920235

02 1 16502899 1650(165871)

2 5446146253 114051623

CC 0 1 223732854 22375 223733(224499)

2 61672823 6167283 120903392 1209034

01 1 23479607 2361(236144)

2 647210683 12687804

02 1 24563418 2513(248548)

2 677047553 13272398

or

a1=a2=a3=a4=b1=b2=b3=b4=a, (45b)

which will be called cubic and quartic beams respectively. For these beams, the first naturalfrequency also remains fairly constant over the range of values considered for the parametera. The accuracy of the present approach is shown in Table 6 for cantilever beams withcross-sections varying as in equation (45a) with a=1. The full RayleighRitz analysis isshown to converge rapidly, and the one-term RayleighRitz approximation has only 029%error.

Mabie and Rodgers [8] obtained exact solutions for the vibration of beams of rectangularcross-section with constant width and linearly varying height. Recently, Cortinez andLaura [15] presented a one-term approximation for that problem. In that case, the onlynon-zero coefficients in equations (20) are

a1=3a, a2=3a2, a3=a3, b1=a, (46)

715

where a is a parameter taken between 1 and 1. Figure 2 shows that the one-termapproximation is very accurate for small values of a. Results for clampedclamped andclampedpinned beams are in good agreement with the results of Cortinez and Laura [15](Table 7).

7. CONCLUSIONS

We have shown that there is a class of non-uniform rods for which the equation ofmotion can be transformed into the wave equation. When both ends of the rod are fixed,the natural frequencies of such non-uniform rods are equal to that of a uniform rod. Forfixedfree and freefree non-uniform rods, the natural frequencies are determined by solvinga simple transcendental equation. For these two cases, the lowest natural frequencies areshown to be affected by the tapering of the rod. However, the higher modes are quiteinsensitive to tapering. In all cases, including the fixedfixed rods, the modes are affectedby tapering.

A class of tapered beams has been shown to allow for transformation of the equation ofmotion into the equation of motion for uniform beams. For clampedclamped beams, thenatural frequencies of all those tapered beams are shown to be always equal to the naturalfrequencies of a uniform beam. For other boundary conditions, the natural frequencies areobtained by setting a 44 determinant equal to zero.

The vibration of a beam of general shape have also been studied using the RayleighRitzapproach. The variation of the cross-sectional area and the moment of inertia are bothassumed to be represented by a polynomial of degree m. Results have been given forseveral cases with combinations of boundary conditions. One-term approximations for thefundamental natural frequency developed here have been shown to be accurate for smalltapers. In addition, we have shown that for beams with either clampedclamped orspinnedpinned ends and the same polynomial variation of cross-section and momentof inertia, the first natural frequency remains nearly constant as the taper parameter a variesover a wide range.

REFERENCES

1. M. E 1991 Applied Acoustics 34, 123130. Exact longitudinal vibration frequencies ofa variable cross-section rod.

2. K. F. G 1975 Wave Motion in Elastic Solids. Columbus, Ohio: Ohio State University Press.3. J. W. M and J. P. W 1992 Journal of Geotechnical Engineering 118, 667685. Cone models

for homogeneous soil. I.4. J. W. M and J. P. W 1992 Journal of Geotechnical Engineering 118, 686703. Cone models

for soil layer on rigid rock. II.5. J. W. M and J. P. W 1993 Earthquake Engineering and Structural Dynamics 22, 759771.

Why cone models can represent the elastic half-space.6. S. A 1993 Final Report for Summer Faculty Research Program, Air Force Office of Scientific

Research, 15.115.19. Wave propagation during high velocity impact on composite materials.7. H. D. C, E. C. H. B and J. F. D 1964 Transactions of the American Society of

Mechanical Engineers, Journal of Applied Mechanics 31, 329331. Vibration frequencies of taperedbars and circular plates.

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12. W. L. C and P. J 1993 Journal of Sound and Vibration 166, 521529. Transversevibrations of a linearly tapered cantilever beam with constraining springs.

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extension of Timoshenkos method and its application to buckling and vibration problems.16. D. H. H, Y. Y. C and X. Y. S 1994 Journal of Sound and Vibration 169, 276283.

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Prentice-Hall.

APPENDIX A: APPROXIMATION FUNCTIONS, STIFFNESS AND MASSCOEFFICIENTS FOR ONE-TERM APPROXIMATION

f=x(x1)(1+xx2),

k=EI0(168+84a1+48a2+30a3+20a4)/35L3,

m=rA0L(8866+4433b1+2509b2+1547b3+1016b4)/180 180.

f=6x24x3+x4,

k=EI0(1008+168a1+48a2+18a3+8a4)/35L,

m=rA0L(416 416+334 048a1+278 356a2+238 329a3+208 236a4)/180 180.

f=x2(x1)(123x),

k=EI0(252+105a1+66a2+45a3+32a4)/315L3,

m=rA0L(10 868+6169a1+3770a2+2457a3+1680a4)/3 243 240.

f=x2(x1)2,

k=EI0(84+42a1+32a2+27a3+24a4)/105L3,

m=rA0L(572+286a1+156a2+91a3+56a4)/360 360.