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A Note on the Coupled Free Bending and Torsional Vibrations
of Beams
YOSHIDA, Shun-ya
信州大学工学部紀要 7: 21-30 (1957)
A Note on the Coupled Free Bending and
Torsional Vibratioins of Beams
(Department of Civil Engineering, Faculty of Engineering, Shinshu University)
. (Received July, 25, 1957)
Synopsis. In discussing lateral vibrations of beams it is always assumed that
the beam vibrates in its plane of symmetry. If it is not the case, the lateral
vibrations will usually be coupled with torsional vibrations.
This paper deals with the natural vibrations of beams in
axis is not collinear with the centroidal axis. Fundamental
frem energy consideratiens which, in turn, are based upon
deflection curves in bending and torsion. The Rayleigh-Ritz
which the shear-center
expressions are derived
assumed normal elastic
method is employed to
determine the natural circular frequencies. Frequency equations thus obtained
involve some dimensionless values which depend upon the various physical character.
istics and the end conditions of the beam under consideration.
Imtroduction. Consider the natural vibrations of the beam as showR in
Fig. 1 in which the longitudinal axis G whlch passes through the mass centers
of the elementary sections is not collinear with the longitudinal axis C about
which the beam tends to twist under the influence of an applied torsional
This axis C, here we call it a shear-center axis(6', may be defined by
the property that it is the oBly axis along which transverse loads applied to
the beain wlll produce flexure without torsion.
These two axes just described are not usually collinear in most prismatic
beams having nonsymmetrical sections, as well as in built-up beams whose
structurai members are not symmetrically placed.
/t 7
t? '
Type of Beams ln wliich Ceutroidal Axis is not
Collinear with Shear-Center Axis.
Shun-ya YosmDA
No. 7
The normal modes of vibration of such a beam involve simultaneous
displacements iR flexure and torsion. Accordingly, the natural frequencies of
vibration in the several normal modes differ from the frequencies computed
for vibration in pure flexure and pure torsion, respectively. (i)(2)(3)(4)(5)
These modes of vibratien are discussed by Timoshenko(i), Clyne F.
Garlancl(7) and the others(8), but it'is assumed that the flexural rigidity of the
beam, say, in the x-y plane is very much greater than that in the y-2
plane. Thus, the x component of the motion is neglected and the total
motion is considered to be composed of the z and 0 components. This
assumption may be reasonable when the beams'are much stiffer in one plane than
in the other. But in general types of beam as used in rigid frames, bridge trusses
etc., it is considered that the above assumption may not always be reasonable.
In the following discussion, the writer deals with beams which may be
deflected in arbitrary direct'ions.
The author is heartily grateful to professor Kinzi Shinohara of the Kyushu
University 'for his invaluable advices and encouragements in the coHrse of this
(1) Tke Expressions for Energies
In the free vibrations o'f a beam of the type shown in Fig.1, the displacement of any section is considered to be the resultant of the following
three components.
2 =:- i・- (at Yi -i- a2 Y,, -F a3 Y3 + ・・・+・・・・・) Sin Pnt, )
0==" (9Pi Yi' "F 9L, Y2' ti- S03 Yh "'l- '''''') Sin Pnt, t ''"'""-・-"'''''`'''・・・・・・・・(1)
Jv =:-, (bi Yi `' + bL・ Yl., `' -i- b3 IKi `'+ ・・+・t・) sin Pnt, 1
where z denotes the vertical component of displacement at any section; 0
denotes the angular displacement at any section; x denotes the horizontal
component of displacement at any section; Yi, Y2,・・・・・-, Y'i, Y'2,・・h・・・, YK,
YM2, ・・・・・- are functions of y which satisfy the end conditions for any particular
beam; and ab a2, a3,・+・・・・, gb g2, g3,・・+・・・, bi, b,o, b3,・・・・・・ are the amplitudes of the respective functions Yot.
As the energies of vibration we take potential energy anct kinetic energy
relating to bending and twisting of the beam, fleglecting the effects of axial
tension, compression or shear.
Then the potential eneirgy of the beam may be expressed as the sum of
the energies stored in the beam due te the displacements z, 0, x, respectively,
v==r}K,S3(Z・l]i;4;,)2dy-FicSg(Z!,2)2dy+±K.Si,(3-tr,le.)2dy, ・・・・・・・・・・・4・・・・・・..,(2)
where V is the total potential energy stored in the beam, Klt being the
flexural rigidity about x axis, K2 being the flexural rigidity about z axis, C
being the torsional rigidity and l being the length of the beam.
Substituting expressions (1) into expression (2), the potential energy is
No.7 ANote on the Coupled Free Bending and Torsional 23
Vibrations of Beams
wrltten as
where Qi, Rt and si denote the integra16 S8(-・4-d2-t-Y-2--l-)2dy, Sg(ddYyt')dy and
S:(4Zt,'")2dy respectively.
Or still more briefly
V :.-5- fi sin2pnt, ・・--・・・・・・・-・・・・・・・・・・・・・・・・・・・・・・・-+-・・・`・・・・・・-・・・・・・-・・・・-・・・・・・・・(2b)
where B denotes the quantity within the brace in equation (2a).
The kinetic energy of the e}ement is expressed as the sum of the kinetic
energy due to the translation of the mass center and that due to the rotation
a'oout the mass center.
Thus, integrating over the length of the beam, the expression for the total
kinetic energy becomes
T == s-,As62zd,+s-,i.sgo2d,-i-s-, Asgdrkd,, ・・・・・・・・・・・-・・・・・・・-・・・・・・・・・.(3)
where T is the total kinetic energy at the displacement 2, 0 and x, p being
the mass density of the material of which the beam is 'made, A being the
area of the beam, 2G and drG being z component and x component of the velocity of the mass center, 0 being the iRstantaneous angular
velocity of rotation of the section and fa being the polar moment of inertia of the section with respect to the gravity axis.
For small values of 0
za=:z+exO, 1
xG :t+e.o. f ''"''''''"''''''"'`''''''''''''''''''''''''''''''''''''''''"'''-"(4)
Hence, by differentiaging equation (4) with respect to time
2a=:-,2+exO, l
rka===fu+e20. f ''''''''''''''''"''''""'-"'''''''''''''-'''''''''''-''''-''`(5)
Substituting values of equation (5) into equation (3) the expression for the
kinetic energy becomes
T =:r5p A[Sg(22+2e.lro'+ ezo2)dy + :-l}-Sgo2dy
sg 'g'
+ (dr2+2ezk0+ee2)dy]. ・・・・・-・・・・t・-・・・・-・・・・・・・・・・・・・・・ny・・・・・・・・・・・・・・・・・(3a)
From equation (1), by differentiation
2=r (aiYi +a2Yh +a3Y3 +・・・-・・・・・・・・) P}2cosPnt, 1
Substituting these values into equation (3a)
t/ t
T ==lyA.{[ke'" af・Ui --22!]aiak Utk] +2exlEiaigkUjfet
+[(ek+e2'+I-A"-)(Eg;UiJ +2Xgic kUi'k')]
ri'["t"Jb?'Ui' +22'bi bh Ut'k' ]tl-2eg -SsoibhUi'k" Pn
l 2cos2Pn t, ''''`''''''''''(3b)
where Ui, Uit, Uint,・・・-・・・・・・・-denote the integrals SoiYl dy, SgYI・2dy,
SgY;・ady, ・・・・・・・・・ respectively and Uik, Uih,, Ui,kv, ・・・・・・・・・・-・・・ denote the
integrals Si,YiYfedN, SgYiY' kdy, IgYl'YZdy,・・・・・・・・・respectively.
Or, more briefly
T==rScrP,7 cos`nd'Pn t, ''''''・+・・・・・・・・・・・・・-・・・・・-・・・Ji・・・・・・・・・・・・・・・i・・・・・・・・・・・・・・(3c)
where cv denotes pA times the quantity within the brace in equation (3b).
Here we apply the Rayieigh-Ritz method to evaiuate the natural frequencies of vibration. Equating the maximum values of the potential and kinetic
energies, as obtained from equations (2b) and {3c), respectively, and solving
for the frequency
p;', ===E. ・・・+・・・・+・・・・・・・+・・・・・・・・・・・・・・・・・・・・・-・・・・・・・・・・・・-・・・・・・・・・・.・.-.・・+・・..<6)
Therefore, the values of Pn2 obtained from equation (6) depend upon the
assumed elastic curves of the beam iB motion. If the assumed elastic curve
is no`t the exact one, the lowest computed value of natural frequency wM be
higher than the true fundamental frequency of the beam. Or, to state it a
little differently, if several elastic curves are assumed in succession, the one
yielding the lowest value of frequency is nearest correct.
In order to obtain the closest approximation possible, the coefficients
ai, a2, ・・・,gi, g2, '''and bi, b2, ・・・must be so chosen that the fundamental frequency
computed from equatioB <6) be a minimum. This value of Pg, rnay be £ound
by equating to zero the partial derivative of Blcu with respect to each of the
coefficients ai, a2,・・・・・・・・・, gi, g2, ・・・・・・・i・and bi, b2,・・・respectively. Thus the
following simttItaneous equations will be obtained.
Z'Pi-r-pgi 3'i",' :== o・ "i/
ao-P.・ell -p?i Z//E --=- o・ 111
----------------------l-----t---- l
b'ip';oB'-- zP'!act
ww O・ Y・・・・・・・・・-・i..・・・......・・・.......・・・・・....・・......(7)
tbe's,-p?tZ-g:,t==e・ i
--t-------------l-l-------l-i-+-- i
3--S-;--p?e ibO:,'i -・ o・ l
No.7 ANote on the Coupled Free Bending and Torsional 25
Vibrations of Beams
g--p,--s・ - p:・, t-6,ny, ., o,
・---・--'・-:H'"'"'""' j
Tliese equations are seen to be homogeneous and linear in the coefficients
ai, a2,・・-'・・ gi, g2, -・・・-・ and bi, b2, ・・・・・・, and are equal in number to the
number of coefficients. By setting the determinant of these equations equal
to zero, eliminating the coefficients, and expanding, the frequency equation
may be obtained. The frequency equation yields the values of the naeural
frequencles in the several normal modes.
(2) The Freq"ency Equatioll for Three Normal wrodes.
In applying the same methed as used in £his ana}ysis to the determi(1)
nation o'f the natural frequencies of a vibrating string, Timoshenl<o shows
us that when only one term is used to express the assumed elastic curve,
the computed value for the fundamental frequency differs from the exact
vaiue by e. 66 per cent. In another example ・・・・・・ ehe case of a vibrating wedge
'of constant width, with the thick end built in and the other end free ・・・・・・
Timoshenl<o also demonstrates that the error in the computed value of the
fundameneal frequency is about 3 per cent when only one term is used to
express the elastic curve. Therefore i# is seen that the method glves very
satisfactory results, even when only one term is used. Moreover, it should
be recognized that when a larger number of germs is used, the extra labour
of computation may not be worth the increased accuracy which uxill be
gained. Thus we express the instantaneous displacement of the beam by the
z = al Yl sin Pnt, )
e= giYi' sin Pnt, ll・ ・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・-・・・・・・・・-・・・・・・t・・・・・・・・(lb)
x=bi Yi" sin Pnt. fl'
From equation (2b)
P:=:.ILva,2Qi--yCgi2.Riff-・Klebi2Si, ・・・・・・+・・・・・・・・・・・・・・-・・・・・・・・・・・・・+・・・・・+・+・・・・(s)
and from equation (3c)
a==pA[di2Ui+2escaigiUll,+ (ex2+e22+LAG-) gi2Uir
--P bi2Ui" -l- 2eznbi Ui ,i tt] ・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・+・i・(9)
Taking partial derivatives with respect to ai, gi and bi, respectively, and
rp-={;i'L', r== <ZliS ¢-C.(--i-f, g=: l.Z- l'i", ・・・・・・・・・・・+・・・・・・・-・・・・・・・・・・・・・・・・(iO)
the following expressions are obtained.
21SI===2K}tQiai, bOpaP-, !2CRigi, tsBb・E-2KhSibi, g-l-l=:2pAU,(a,+exrg,),
oO-S-,pt-2pAUi Cexrai+ezg"bi+ (e.2-Fe.2+k--g--)mp,),
Go-t6I ::,2RAUi (ezk"{ i-l-ipbi).
Substituting these terms into equation (7) and rearranging into determinant £orm
ai -2pAUiexrP"2gi O
l. ---2pAUiexrPn2ai 2(CI?i-pAUi(ex2+ea2+-Il-G・-)pn2n)gi -2pAUieaepn2bi ii.1.(Oit)
i O -2pAUieaePn2gi 2(KleSi-pAUidiPn2)bi
Dividing the first column by 2pAUiai, the second column by 2pAUigi and
the third colurnn by 2pAUibi, the determinant is reduced to the form
tt'fiLQu'i'Iptpn2 pmexrpn2 o l
pCARtii,'--"(ex2+e22+f')vp.2 ,""'O・''''''''''''''''''''''''(11a)
o -e.cp.2 ,",KletSl -{bp.2 1
Now the quantity [pKA]tgeu--i-]2 expresses the natural frequency of flexural
vibration for a uniform beam. This quantity is therefore the natural frequen-
cy of the beam under consideration for the particular case in which ex and
ea are zeros. Then the natural frequency for this particular case may be
denoted by Pe and the relationship
e K}vQi
PO-:="iA'UI 'M''''`'''''''''''''-''''''+・・・・・・-・・・i・・・・・・-・・・・・-・・・・・・・・・+・・・・(12)
may be substituted into equation (11a) and each term in the determin.ant
divided by P,2.
After this operation is performed and the determinant is expanded,
simplified and rearranged, the following cubic equation is obtained.
[ex2r2¢ + ez2s2 - (ex2-l-e22+ I.・4-g) rpip](/tti)3+CE- l-RQ-il ¢- fl}3-41 ex2r2-ez2'g2
+ (ex2+ez2+ig) filSd-io ](/t'i) +gt-・{t519s - o・ ・・・・・・・・・・・・・・・・・・・・・-・・・・・・・・・・・・・・・・・・・・・・・(i3)
Furthermore we introduce the following additional dimensionless quantlties
2=E- -RQ--t'iil!;-?, Ex==t fat2A, ex =:: tt:-.A, rc =l} JiEioSl. ・-・'''・''・'''''''''''''''''''''''''''''(i4)
After all the frequency equation is
No.7 ANote on the Coupled Free Bending and Torsional 27
Vibrations of Beams
[1+ex (1--- li'1) +Ea (1-- S- -,27)] (2p{,'-,2-)3-- C1+1-{-rc --Ex (1+ic-Z;Pl ,c )
+e,(i+rc-ii-2ii)](tt)2 + (a+rc)R --}- (i -i-s.+e.)rc](Pp-'g,2)- irc = o, ・・・+・・・・・・-"s)
and its three roots will be readily obtained as follows.
Let Ai :=[1+ex(1-/2)+e2(1-Sl7i)], . ..
A2F:- (1+R+rc -rex (1+t;-r-rp2 rc)+ez (1+rc--t/i )] ,
A3 =='[(1+rc)ZkF (l-FEx+Ez)rc 1',
A, ==-, -Zrc,
P ="gALs(- A22+3AiA3),
1 (2A,3---9A,A,A,+27A,2A,).
q ==54A,3
Case (i): when q2 +p3>O.
;tet==u+v----3AA21, 2p-:i==ua1+vto2--glAIk, Sg/lg==uto2+ve)1---3AA21,
where u=ntY=q' +Vq2+p3, V=er-q-Vq2+fi}ii,.
Case (ii) : when q2+p3 ==O.
Silill22=:' 2 iil/ ii':"e - gA'ln2,, iili;'--'illi;2E=:-ii}/:rmLq-・
Case (ii2) : when q2+p3 < e.
$:i--- 2,.・t' - p cos (l; -i- -{l'k) - 3AA2, ・
where cos u=t-q:--=,..:w. , when O<u<T.
pv- p
(3) The Frequeney Equations for Special Cases
(i) When the beam has one plane of symmetry.
For instance, when the cross section of the beam is symmetrical about
x-axis (Fig.2), put ez:=tO in equation (15). Then tlte frequency equation is
--i-C(1+rc)2--(1-I-ex)tg](ge,;)-2rc ntO. ・・・・・・・・・・・・・-・・・・・・・-・・・・・・・-・・・・・・・・・(16)
(ii) In case (i), when K2 is very much greater than Kte.
Dividing equation (16) by rc and putting ti----・co
[1-l-ex(1-7-,'72 ))t31)L'-a+R-l-ex) (Pp-',-'i) +Jl==-,e. ・・・・・・・・・・+・-・・・・・・・-・・・・・・(17)
This equation is the same one as has been obtained by Garland.
(iii) When the shear-center axis is collinear with the ceRtroidal axis (Fig.3).
Putting ex==,c-xt=tO in equation (15)
(;.{/1"i.)3-(1+ft--i-rc)(lp)',t.]-)2' -{-C(1+rc)R-i-fi;](SLIg) -2-J ==:O. ・・''''"''''''''''J(18)
Fig. 2. Fig. 3.
(4) Diseasssgoms about tke Fyequencies
It is noted that the three roots of equation (18) are 1.0, 2 and rc respectively,
This is explained by the fact that in the actual beam, if the shearcenter axis is collinear with the centroidal axis, two of the normal modes of
vibration are those of pure flexure, and the other one is that of pure torsion.
Thus, it will' .be inferred that the frequeBcies of a beam in which the effects of
eccengricity are not neglected differ from those in pure flexure or in pure
torsioR. That is to say, the natural frequencies of beams in which the shearcenter axis is not collinear with the centroidal axis depend upon the distance
of these two axe$. This will be shown in a numerical example as fo}lows.
Omitting the process of calculations, only the results are presented graphycally in Fig, 4.
(5) Smammary
The natural frequencies of the beam in which the shear-center axis is
not collinear with the centroidal axis are shown to differ from those in pure
flexural or pure torsional vibrations, and the normal mode of vibration of
this beam consists of simuleaneous vibrations in flexure and torsion. Thus,
A Note on the Coupled Free Bending and Tor$ional
Vibrations of Beams
No. 7
Fig. 4. Relations between pnlpe and Ez! c-x wken
rp =1 ,r= O.8, rc=2. Numbers on Curves
Denote Values o£ 2.
1. 2
1) iv
it is seen that computations oE £he natural frequencies of such a beam, in
which the effects of the eccengricity are neglected, are apt to lead eroneous
When the type of beam, end conditions and load distribution are known,
the values of natural frequency of the beam can be obtained from the frequency equation (15). Kigher degree of accuracy will be attained by using a
sufficient number of terms in the expressions (1>, but, as mentioned previously, for most practical problems satisfactory values of frequency may be
obtained by using only one or two terms. The absolute amplitudes are of
course arbitrary since they depend ttpen the initial displacement of the beam,
but the amplitttde ratios will be found by bubstituting the values of natural
frequency obtained from the freqttency equation into equation (11).
(1) Vibration problems in Engineering, by S. Timoshenko, D.van Nostrand Company,
Inc., New Yerk, third editien, 1955.
(2) Shind6-Gaku, by Katsutada Sezawa, Iwanami Book Company, Tekyo, sixth edition,
(3) IV[echanics Applied to Vibrations and Balancing, by D. Laugharne Thornton, Chap-
man & Hall LTD., Lonclon, second edition, 1951.
(4> Theory of Vibrations for Engineers, by E.B. Cole, Crosby Lockwood & Son LTD.,
London, second edition, 1950.
(5) Approximate Calculations of the Vibration ef Plate Girder Bridges. Freqttency of
Natural Vibration due to Bending only, by Kinzi Shinohara, Memoirs o'£ The College
ofEngineering, Kyushu Imperial University, Vol.3, No.16, 1941.
(6) Strength of Materials,by S. Timoshenko, D.van Nostrand Company, Inc., New
Yorl<, 1955, part 1.
(7) Tlie Normal Modes of Vibrations of Bearns Having Noncollinear Elastic and Mass
Axes, by Clyne F. Garland, Journal of Applied Mechanics, September, 1940.
(8) Shind6-Gaku, by Yoshiyulci Yamamoto, Oy6-Rikigaktt K6za, 2-B, Ky6ritsu Book
Company LTD., Tokyo, 195r.