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Vibration of a variable cross-section beam
Mehmet Cem Ece, Metin Aydogdu
*
, Vedat Taskin
Department of Mechanical Engineering, Trakya University, 22030 Edirne, Turkey
Available online 27 June 2006
Abstract
Vibration of an isotropic beam which has a variable cross-section is investigated. Governing equation is reduced to an
ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying
width. Analytical solutions of the vibration of the beam are obtained for three different types of boundary conditions asso-
ciated with simply supported, clamped and free ends. Natural frequencies and mode shapes are determined for each set of
boundary conditions. Results show that the non-uniformity in the cross-section influences the natural frequencies and the
mode shapes. Amplitude of vibrations is increased for widening beams while it is decreased for narrowing beams.
Ó2006 Elsevier Ltd. All rights reserved.
Keywords: Beam; Variable cross-section; Vibration; Analytical solution
1. Introduction
Beams are used as structural component in many engineering applications and a large number of studies can
be found in literature about transverse vibration of uniform isotropic beams (Gorman, 1975). Non-uniform
beams may provide a better or more suitable distribution of mass and strength than uniform beams and there-
fore can meet special functional requirements in architecture, robotics, aeronautics and other innovative engi-
neering applications and they have been the subject of numerous studies. Cranch and Adler (1956) presented
the closed-form solutions (in terms of the Bessel functions and/or power series) for the natural frequencies and
mode shapes of the unconstrained non-uniform beams with four kinds of rectangular cross-sections. Similar
closed-form solutions for the truncated-cone beams and the truncated-wedge beams were obtained by Conway
and Dubil (1965).Heidebrecht (1967) determined the approximate natural frequencies and mode shapes of a
non-uniform simply supported beam from the frequency equation using a Fourier sine series. Branch (1968)
optimized fundamental frequency of transverse oscillation of beams with variable cross-section which is
allowed to vary in a manner such that the second area moment is linearly related to the area. Mabie and Rogers
(1972) considered polynomial variation of the beam cross-sectional area and the moment of inertia and
obtained natural frequencies for a double-tapered beam. Bailey (1978) numerically solved the frequency equa-
tion derived from the Hamilton’s law to obtain the natural frequencies of the non-uniform cantilever beams.
0093-6413/$ - see front matter Ó2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mechrescom.2006.06.005
*
Corresponding author. Tel.: +90 284 225 1395; fax: +90 284 212 6067.
E-mail address: metina@trakya.edu.tr (M. Aydogdu).
Mechanics Research Communications 34 (2007) 78–84
www.elsevier.com/locate/mechrescom
MECHANICS
RESEARCH COMMUNICATIONS
Olhoff and Parbery (1984) used cross-sectional area function as the design variable to maximize the difference
between two adjacent natural frequencies. Gupta (1985) numerically determined the natural frequencies and
mode shapes of the tapered beams using a finite element method. Jategaonkar and Chehil (1989) studied
non-uniform beams with cross-section varying in a continuous or non-continuous manner along their lengths.
Naguleswaran (1992, 1994a) determined the approximate natural frequencies of the single-tapered beams and
double-tapered beams with a direct solution of the mode shape equation based on the Frobenius method.
Naguleswaran (1994b) also investigated a uniform beam of rectangular cross-section one side of which varies
as the square root of the axial coordinate. Laura et al. (1996) used approximate numerical approaches to deter-
mine the natural frequencies of Bernoulli beams with constant width and bilinearly varying thickness. Datta
and Sil (1996) numerically determined the natural frequencies of cantilever beams with constant width and lin-
early varying depth. Caruntu (2000) examined the nonlinear vibrations of beams with rectangular cross section
and parabolic thickness variation. Recently, Elishakoff and Johnson (2005) investigated the vibration problem
of a beam which has axially non-uniform material properties. Free vibration of stepped beams has also received
a considerable attention and a comprehensive review is given by Jang and Bert (1989a,b). Some of these results
can also be found in the monograph by Elishakoff (2005).
Previous studies clearly show that vibration characteristics of isotropic beams with continuously changing
cross-section have significant features and are not yet fully addressed. The present study investigates free
vibration of an isotropic beam with exponentially varying width. The object is to obtain analytical solutions
describing the vibration behavior of the beam under different boundary conditions and to determine the effects
of continuously variable cross-section on the natural frequencies and mode shapes.
2. Analysis
Consider an isotropic beam with a variable cross-section. Dimensionless variables are defined according to
t¼1
L2ffiffiffiffiffiffiffiffi
EI
0
qA
0
st;x¼x
L;I¼I
I
0
;w¼w
W;A¼A
A
0
;ð1Þ
where t
*
is the dimensional time, x
*
is the dimensional coordinate measured from the left end of the beam
along its length, A
*
and I
*
are the dimensional area and moment of inertia of the cross-section of the beam
respectively, w
*
is the dimensional transverse displacement, qis the mass density per unit are of the
beam, Eis the Young’s modulus, Lis the length of the beam, Wis any reference displacement and A
0and I
0
are respectively the area and moment of inertia of the cross-section of the beam at the left end of the beam where
x= 0 that is A
0¼A
0ð0Þ;I
0¼I
0ð0Þ. Governing equation in the dimensionless form can be written as follows:
IðxÞ
AðxÞ
o4w
ox4þ2I0ðxÞ
AðxÞ
o3w
ox3þI00ðxÞ
AðxÞ
o2w
ox2þo2w
ot2¼0:ð2Þ
Solution of the Eq. (2) can be assumed in the following form:
wðx;tÞ¼FðxÞGðtÞ:ð3Þ
Substitution of Eq. (3) into Eq. (2) yields two ordinary differential equations.
IðxÞ
AðxÞFð4Þþ2I0ðxÞ
AðxÞF000 þI00ðxÞ
AðxÞF00 x2F¼0;ð4Þ
G00 þx2G¼0:ð5Þ
Here xis a real constant and defined as x
2
=(X
2
qL
4
/EI
0
)andXis radial frequency. Solution of Eq. (5) is well
known and can be written as
GðtÞ¼C1cosðxtÞþC2sinðxtÞ:ð6Þ
Solution of Eq. (4) requires the geometry of the cross-section of the beam to be specified. A cross-section
geometry for which both the area and the moment of inertia are directly proportional to the characteristic
width is considered in the present study. The family of the cross-section geometries possessing these features
M.C. Ece et al. / Mechanics Research Communications 34 (2007) 78–84 79
includes rectangle and ellipse and the others can be found in the tables given by Hibbeler (2001). Furthermore
the characteristic height of the cross-section or the thickness of the beam is kept constant and the characteristic
width of the cross-section is assumed to vary exponentially along the length of the beam so that A(x)=e
dx
and
I(x)=e
dx
. Here dis the non-uniformity parameter.
For the family of the cross-sections with exponentially varying characteristic width and constant character-
istic height, Eq. (4) reduces to:
Fð4Þþ2dF000 þd2F00 x2F¼0:ð7Þ
Solution of Eq. (7) can be obtained as:
FðxÞ¼ed
2x½B1cosðk1xÞþB2sinðk1xÞþB3coshðk2xÞþB4sinhðk2xÞ;ð8Þ
where,
k1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4xd2
p2;k2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4xþd2
p2:ð9Þ
In the present study, the ends of the beam were considered to be simply supported (S) or clamped (C) or free
(F). The boundary conditions associated with both ends being simply supported (SS), both ends being
clamped (CC) and the left end being clamped while the right end being free (CF) may be written in the same
order as:
Fð0Þ¼0;F00ð0Þ¼0;Fð1Þ¼0;F00 ð1Þ¼0;ð10Þ
Fð0Þ¼0;F0ð0Þ¼0;Fð1Þ¼0;F0ð1Þ¼0;ð11Þ
Fð0Þ¼0;F0ð0Þ¼0;F00ð1Þ¼0;F000 ð1Þ¼0:ð12Þ
3. Solutions
Solution of Eq. (7) subjected to either one of the boundary conditions given by Eqs. (10)–(12) can be writ-
ten as:
FðxÞ¼B2ed
2x½b1cosðk1xÞþsinðk1xÞb1coshðk2xÞþb4sinhðk2xÞ:ð13Þ
Here the coefficients b
1
and b
2
depend on dand x. Application of the boundary conditions in each case yields
an implicit equation for the determination of the natural frequency xfor a given non-uniformity parameter d.
The coefficients b
1
and b
4
and the natural frequency equation are given below for each physical case considered
in the present study.
Case I: Both ends of the beam are simply supported (SS).
b1¼ dð2k1sinh k22k2sin k1Þ
dk2ð2 cosh k22 cos k1Þþxsinh k2
;ð14Þ
b4¼k1
k2
2x
dk2
b1;ð15Þ
4d2k1k2cosh k2cos k1þð8x2d4Þsinh k2sin k14d2k1k2¼0:ð16Þ
Case II: Both ends of the beam are clamped (CC).
b1¼k1sinh k2k2sin k1
k2ðcosh k2cos k1Þ;ð17Þ
b4¼k1
k2
;ð18Þ
4k1k2cosh k2cos k1d2sinh k2sin k14k1k2¼0:ð19Þ
80 M.C. Ece et al. / Mechanics Research Communications 34 (2007) 78–84
Case III: The left end of the beam is clamped while the right end is free (CF).
b1¼2k1ð2dk2d22xÞe2k24k2½2dk1cos k1þð2xd2Þsin k1ek2þk1ð2k2þdÞ2
k2f2ð2dk2d22xÞe2k2þ4½ðd22xÞcos k1þ2dk1sin k1ek2ð2k2þdÞ2g;ð20Þ
b4¼k1
k2
;ð21Þ
2k1f16ðk2dÞx2e2k2ð2k2þdÞ2½2ðd22xÞk2d3g cos k1þd½8ð3d4k2Þx2e2k2þð2k2þdÞ2
ð8k2
1k2d3þ2dxÞ sin k1þ4k1k2ek2½ð2k2þdÞ2ð2dk2þd2þ2xÞþ8x2¼0. ð22Þ
The unsteady transverse vibration of the beam can then be written as
wðx;tÞ¼ed
2x½b1cosðk1xÞþsinðk1xÞb1coshðk2xÞþb4sinhðk2xÞ½c1cosðxtÞþc2sinðxtÞ:ð23Þ
4. Results
The analysis presented describes the free transverse vibration behavior of a beam with exponentially vary-
ing characteristic width and provides analytical solutions. The natural frequencies for the SS, CC and CF
Fig. 1. Frequency variation for a beam with exponentially varying width under the SS conditions.
Fig. 2. Frequency variation for a beam with exponentially varying width under the CC conditions.
M.C. Ece et al. / Mechanics Research Communications 34 (2007) 78–84 81
boundary conditions were obtained by numerically solving the implicit Eqs. (16), (19) and (22) respectively.
The Newton method was used in finding the roots of these equation in terms of the natural frequency x
for a given non-uniformity parameter dand the iterations were stopped when the absolute value of the differ-
ence of the natural frequency obtained at two successive iterations was less than 10
5
. Variations of the trans-
verse vibration natural frequency ratios of a non-uniform beam with exponentially varying width with the
non-uniformity parameter dare shown in Figs. 1–3 for the SS, CC and CF boundary conditions respectively
where x
0
is the natural frequency of the uniform beam. The natural frequencies were also listed in Table 1.It
may be noted that the natural frequencies ratios for the SS and CC boundary conditions are independent from
the sign of dsince the implicit equations for the natural frequency involve d
2
only. All the natural frequencies
of the non-uniform beam are greater than those of the uniform beam for the CC boundary conditions and the
frequency ratios increase with the non-uniformity parameter dand decrease with the mode number. The fun-
damental natural frequency of the non-uniform beam for the SS boundary conditions is observed to be
decreasing with the non-uniformity parameter dwhile the higher frequencies are increasing. The frequency
Fig. 3. Frequency variation for a beam with exponentially varying width under the CF conditions.
Table 1
Natural frequencies for a non-uniform beam with exponential width variation
jdjMode number Natural frequencies
SS CC CF
0 1 9.86960 22.37327 3.51602
2 39.47841 61.67281 22.03449
3 88.82643 120.90338 61.69721
4 157.91367 199.85945 120.90191
5 246.74011 298.55552 199.85953
(d<0) (d>0)
1 1 9.77291 22.51167 4.72298 2.85833
2 39.57036 61.85968 24.20168 20.03917
3 88.97052 121.10799 63.86448 59.87084
4 158.08418 200.07411 123.09790 119.09862
5 246.92650 298.77661 202.06876 198.06964
2 1 9.48725 22.93771 6.25877 2.90893
2 39.85231 62.42272 26.58350 18.17520
3 89.40520 121.72272 66.37449 58.38868
4 158.59689 200.71860 125.68471 117.69217
5 247.48629 299.44012 204.69531 196.70224
82 M.C. Ece et al. / Mechanics Research Communications 34 (2007) 78–84
ratios for the higher frequencies ratios increase with the non-uniformity parameter dand decrease with the
mode number similar to the case associated with the CC boundary conditions. All the natural frequencies
of an exponentially narrowing beam are greater than those of the uniform beam for the CF boundary condi-
tions and increase with the increasing magnitude of the non-uniformity parameter dsuch that the frequency
ratio is smaller for higher mode numbers. But the natural frequencies of an exponentially widening beam on
the other hand are smaller than those of the uniform beam for the CF boundary conditions with the frequency
ratios being higher for higher mode numbers. Furthermore, the fundamental natural frequency first decreases
and then starts increasing with the non-uniformity parameter dwhile the higher frequencies all decrease with
it. It can be easily shown that the last two terms in Eq. (13) behaves like (b
4
b
1
)e
xx
for finite xand large x.
The results also that the difference (b
4
b
1
) vanish with increasing values of the natural frequency leaving the
two trigonometric functions dominant as in the case of a uniform beam. Therefore the difference between the
corresponding natural frequencies of the uniform and non-uniform beams decreases with increasing mode
numbers and the non-uniformity is more pronounced for lower modes. The natural frequencies obtained in
the present study for a uniform beam are in excellent agreement with the results found in the published liter-
ature (Blevins, 1984). The natural frequencies obtained in the present study for an exponentially narrow-
ing beam (d=1) are listed in Table 2 in comparison with those obtained by Cranch and Adler (1956)
Table 2
Natural frequencies for an exponentially narrowing beam (d=1) under the CF conditions
Mode number Cranch and Adler (1956) Tong and Tabarrok (1995) Present study
1 4.735 4.7347 4.72298
2 24.2025 24.2005 24.20168
3 63.85 63.8608 63.86448
4 – 123.091 123.09790
5 – – 202.06870
Fig. 4. First (a), second (b) and fifth (c) mode shapes for the cases considered.
M.C. Ece et al. / Mechanics Research Communications 34 (2007) 78–84 83
approximately with series expansion and by Tong and Tabarrok (1995) numerically. The agreement is consid-
ered to be excellent.
First, second and fifth mode shapes for the cases considered in the present study are shown in Fig. 4. It may
be seen from Eq. (23) that the amplitude of the transverse vibrations is proportional to ed
2x. Therefore ampli-
tude of the mode shapes for a given non-uniformity parameter dincreases with xfor narrowing beams (d<0)
and decreases with xfor widening beams (d> 0). Amplitude is also amplified in the case of a narrowing beam
(d< 0) since less mass per unit length is involved while it is suppressed in the case of widening beam (d>0)
which has more mass per unit length than a uniform beam. Non-uniformity slightly changes the position of
the points and this may provide some design flexibility in engineering applications.
5. Conclusions
The present study deals with the free vibration of non-uniform cross-sectional isotropic beams. Euler–Ber-
noulli beam theory was used in the analysis. Variation of width of the beam was chosen exponentially. It is
found that frequencies are independent from exponential decrease or increase but mode shapes are affected
by the increase or decrease behavior. This study can be extended to other structural members i.e. plates
and shells or to other materials composites or functionally graded materials.
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