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A new method for the vibration of thin-walled beams
0~mput.r~ Printed
& Struchres
in
Vol.39.
No. 6, pi.
597601,
00457949/91 s3.w + 0.00 Pergamon Press pk
1991
GreatBritain.
A NEW METHOD FOR THE VIBRATION THIN-WALLED BEAMS
OF
ZONGFEN ZHANG and &JHUAN CHEN
Department of Mechanics, Jilin University of Technology, Changchun, People’s Republic of China (Reeeiued 19 April
1990)
Ahstraet-This paper presents a new method for thin-walled beams with constrained torsional vibration. Based on the differential equation for torsional vibration, which includes the effect of cross-sectional warping, the shape functions are determined and, in turn, the frequency-dependent mass and stiffness matrices are derived. As an application of the new method, the dynamic finite element method, a numerical example is presented. The results show that this method gives more accurate results in the high-frequency range than those obtained by the static finite-element method.
NOMENCLATURE
Shape function modulus of elasticity indicating element modufus of rigidity (or shear m~ulus) polar moment of inertia of cross-section torsional constant warping constant stiffness matrix mass density mass matrix kinetic energy eigenvector displacement vector velocity vector potential energy angle of twist of beam at any cross-section eigenvalue u 0
Poisson rate natural frequency
INTRODUCTION
Thin-walled beams are wildly used in a broad range of structural applications, such as automobile and aircraft frames. Studies involving static or dynamic calculations of thin-walled beams have been completed by many authors, including Gere [l], Bathe and Chaudhary[2] and George [3]. The common features of these studies is that the static displacement interpolation functions are used for the dynamic analysis. The eigenvalues obtained using the corresponding mass and stiffness matrices are unreliable in the high frequency range. In order to increase
accuracy, we have to abandon the traditional static displacement inte~olation method. From the motion equation of elements, we build the displacement interpolating functions which are the functions of unknown frequency. The corresponding dynamic mass and stiffness matrices are derived. This method
is called the dynamic finite element method. To avoid complicated calculations we assume that the displacement can be expressed as an ascending power series of the unknown frequency [4]. Using this kind of displacement, in the published papers, the dynamic mass and stiffness matrices of bar, beam and plate have been derived. But so far the dynamic mass and stiffness matrices of thin-walled beams with constrained torsional vibration have not been determined. From the differential equation this paper derived the dynamic mass and stiffness matrices of the thinwalled beams with constrained torsional vibration. The application of this method is shown by an example. The results show that the dynamic finite element method has a high accuracy. DIFFERENTIAL EQUATIONS OF THIN-WALLED BEAMS WITH CONSTRAINED TORSIONAL ~BRA~ON
The differential equation for free torsional vibrations of thin-walled open or closed cross-section beams is O(4) -
KQ’2’
.+_ p28’
=
0
,
where K2=-,
E, =P =
GJk 4 J,
p2~!!!% &Jo
E
1 -C2’
1,
P==-J&
for open cross-section, for closed cross section.
(1)
598
Z~NGFEN ZHANGand
SUHUAN CHEN
Assume that
O=tiU,
G
(2)
where (3) and
r\
Lr
u, =e1 x=0
u,=e
u2 =@I x=0
u,=WI x=1
1 x=1
Fig. 1. End displacements of the beam with constrained torsional vibrations.
also Ui, i = 1,2,3,4 are the end displacements of the beam shown in Fig. 1. Substituting eqns (3) and (4) into (2), one gets
=Oda,
(5) 1-u
eqn (5) into (l), we obtain
- pz,io @_d+2qeior= 0. ‘_
(6)
The coefficients of the same powers of w in eqn (6) must be zero, therefore we obtain d&” - K2rjh2)= 0,
(7)
a\” - K2Z\2)= 0,
@a)
1.
@b)
Equations (7) and (8) can be integrated directly. The first matrix rI, is then used to satisfy the boundary conditions that 0 = u, and 8 = u, at x = 0, 6 = uj and 8 =u4 at x = 1, while the remaining matrices - _ must all vanish at x = 0 and x = 1. al,a2,a3,..., The general solution of eqn (7) is as follows: G&c) = ?‘, + F2x + F,Shkx + F,Chkx
F,(l) = A/S,
F2(1) = - KShkl/S,
F,(l) = Shkl,
F,(l) = B/S,
4 (2) = DlKlS,
F,(2) = B/S,
f-3(2)= A IKIS,
F4(2) = -D/K/S,
F,(3) = B/S,
F,(3) = KShkl/S,
F,(3) = -Shkl/S,
F,(3) = -B/S,
F, (4) = HIKIS,
F2 (4)
F,(4) = -B/K/S,
F,(4) = -H/K/S,
(lla)
(11’4
(1lc)
= B/S,
(114
a,,(O) = 1, lX_,=O,
S = KlShKl - 2Chkl -I-2, A = KIShkl - Chkl + 1, B= 1 -Chkl, D = klChk1 - Shkl, H = Shkl - ki. As matrix d, must vanish at x=0 according to eqn (8) we obtain
(9)
requiring that eqn (9) satisfy the boundary conditions yields
UOa)
and x=1,
a, - = 0.
(12)
In order to avoid tedious algebraic manipulation, the following procedure is adopted to solve eqn (8b). To this end, (9) can be expressed as follows: a,,i = F, (i) -I-F2(i)x + F, (i)Shkx + F,(i)Chkx,
i = 1,2, 3,4.
aoz(0) = 09
uo2(l)=0,%
&
i,..=l’~i,_,=o’
(104
where
ni4) - K2fii2’ = p2fio,
aO,(l)=O,% dx /,.,-O,z
=
=I
Substituting eqn (9) into (10) the coefficients in (9) can be found as follows:
Note that for simplicity we define
Substituting
’ dx
=0
(lob)
sub;;~;(ek~_e-kX),2,
Chkx+kX+e-““)/2
(13)
599
Vibration of thin-walled beams Equation (19) can be written as
into eqn (13) gives qi = F, (i) + F2(i)x + (&
+ (F4(i) - F,(i))eekX/2. Substituting leads to a2i=
T(t)‘= ;!?TTM’t?.
(i) + F4(i))ek”/2 (14)
eqn (14) into (8b) and integrating
(8b)
Comparing matrix is
eqn (19) with (20), the element mass
M’=
F,(i)+F,(i)x
+F7(i>x2+F8(i)x3
+ Flo(i)e-k”+ F,,(i)x&
+
‘+ir&dx. I0
F9(i)ek”
+ F,#xewk”,
(15)
(20)
Substituting
(21)
eqn (3) into (21) yields I
M’=
where F,(i) = p2F4(i)/K4 + C,(i),
(16a)
Fe(i) = p2F,(i)/K3/2 + C,(i),
(1W
F,(i) = -p*F, (i)/K2/2,
(16~)
F,(i) = -p2F2(i)/K2/6,
(16d)
F,(i) = -p2(F3(i) + F4(i))/K4/2 + C,(i),
(16e)
F,,(i) = -p2(F4(i) - FI(i))/K4/2 + C,(i),
(160
MZJ$ + f&u2 f . * . y I xo(4+d,w’+...)dx
(22) where A4;=
K,(i) = p2(F4W - 4WY~3/2,
iUS =
=
-P'(F~(z)
-
(1W
Wr))/K3/4,
(17a)
C.,(i) =p2(2 + kl)(F,(i) - F,(i))/K4/4,
(17b)
C,(i) = - C,(i) - C,(i),
(17c)
C,(i) = KC,(i) -kc,(i).
(17d)
J
@I”)2dx/2 + GJk - @I’)*dx/2. (25)
0
Substituting
C,(i) =p2(2 - kl)(F,(i) + F4(i))/K4/4,
(24)
Pl
r1
(16h)
where
’rnZ,(ti~lr,+ Crl$) dx.
I0
From the theory of constrained torsional vibration, the element potential energy is V(tY = E*J,
Fl2G)
(23)
‘mIPa&i&t f0
J0
eqn (2) into (25) we obtain
v(ty = E,J,
’ t7T(cY5iW)t7dx/2 I0 I
+ GJ,
I0
ur(a’%‘)t7dx/2.
(26)
Equation (26) can be written as
Using eqns (1 l)-( 17), 4 and 4 calculated on a digital computer. DYNAMIC
can be easily
Substituting
‘mZ,,~2dx
Comparing eqn (26) with (27) we obtain the element stiffness matrix K’ =
This section presents the calculation of the element mass and stiffness matrices. From the theory of constrained torsional vibration, the element kinetic energy is
s0
(27)
1
MASS AND STIFFNESS MATRICES
T(tY=k
V(t) = ; WK~U.
V-9
’ E,&irTcin
s0
dx $
GJ,li’%’ dX. s0
(28)
Substituting eqn (3) into (28) we get K’ =
’ E, J&i; + o*d; + . . . )r s e,,,.,,;....,,,
eqn (2) into (18) we obtain
s
x (zi;+o%;+***)dx
1
T(t)’ =;
mIPtJTTTdTZitrdx.
0
(1%
=K;+o’K;+...
9
(2%
600
ZONGFENZHANGand
where I K;=
E, J,c?;~&’ dx +
s0
GJ,ci;Tci; dx,
(30)
f0 1
K; =
’ E, J,~i;~ci; dx +
I0
GJkci;T~; dx.
following formulas can be used to calculate MO, Mr, &, and k4 I =
cluster of natural frequencies. In order to avoid the difficulties of cluster frequencies, the iterative perturbation (51 for solving the nonlinear eigenproblem eqn (38) can be used. If AM is dropped, eqn (38) becomes
(31)
J0
The
Mi(i,j)
SUHUAN CHEN
mI, aoiaojdx,
k,u = AM,.
(39)
This is the same eigenproblem as obtained by the static finite element method. Equation (38) can be referred as the perturbation equation of eqn (39). Then the perturbed eigensolutions are
(32)
I0
A; = li + Ali,
(40)
I
M;(i, j) =
mlp (a,, a*, +
azta,)
E, J,a%a&dx
+
dx,
(33)
Au;=ui+Aui,
s 0 I
Ko(i, j) = s 0
GJ,a,&aijdx,
(34)
s 0
(41)
where li and ui are the eigensolutions of eqn (39) Ali and Aui can be easily computed by matrix perturbation
I
K;(i, j) =
E, J,ayia$ dx
-rliu;AMui,
Ali=
s 0
(42)
I
GJka&a$dx,
+
i,j=1,2,3,4.
Aui= i
(35)
The matrices rn; and kg represent the static inertia and stiffness of the thin-walled beam element with constrained torsional vibration, while rni and k; and higher-order terms represent dynamic corrections. ITERATIVE PERTURBATION FOR SOLVING NONLINEAR EIGENPROBLEM
In practical calculations, the iterative perturbation starts with eqn (39), and assume that
For P=l,2,...
THE
AM(p) = w#“(M, - K,) - w\f”(M2 - k.,),
The eigenproblem for thin-walled beams with constrained torsional vibration is as follows:
Al(P)
(36)
(k. + L’K, + . . . )u = l(Mo + AM, + . . . )u,
=
where
fq
A =wr.
=
I
(u!P’)TAM(PMP)v i,s=l,2
i = s,
i(~jp))ThW~hjp),
AajP',
(48)
(37) (49) (3W
or
@ + ‘) = #’ + A@‘,
(50) (51)
k, u = 1 (MO+ AM)u,
AM = 1(M, - M4).
(47)
,..,,
Equation (37) can be rewritten as k,u = l[Mo + 1(M, - K4)]u
(46)
I
a!p + ” = alp) +
=1(Mo+1M2)u.
I?
(45)
Iz.!P’
m
i#s,
-
Neglecting the higher-order terms, we arrive at
_~!P)(U!~))TAM(P)u!P) I I
I -
(&++‘K,)u
(43)
c&u,.
s=,
s 0
(38b)
Equation (38) is a nonlinear eigenproblem. The method for solving eqn (38) has been presented by Wittrick and Williams [6], which was based on the sequence search in every neighbourhood of interested frequencies. However, it is difficult to search a
(52) (53)
where .Cis the convergence tolerance. In general, after four to five iterations, convergence can be achieved.
601
Vibration of thin-walled beams Table 1. The comparison of eigenvalues of simple supported beam with constrained torsional vibration 6 Elements
10 Elements
DFEM (error X)
SFEM (error %)
Modal number
Exact solution
SFEM (error %)
DFEM (error %)
1
O.l37116E+6
0.13727OE + 6 (l.l169E1)
0.137116E + 6 (0)
0.138005E + 6 (0.648356)
O.l37118E+6 (0.14586E - 2)
0.564519E + 6
0.567104E + 6 (4.5786E - 1)
0.564524E + 6 (9.317E - 4)
0.579400E + 6 (2.636)
0.564607E + 6 (0.01588)
0.133037E + 7
0.134458E + 7 (1.0678)
0.133044E + 7 (4.735E - 3)
0.141089E + 7 (6.0592)
0.1331728E + 7 (0.1020766)
0.251495E + 7
0.256495E + 7 (1.988)
0.251537E + 7 (1.661268 - 2)
0.278887E + 7 (10.8917)
0.252837E + 7 (0.5336)
0.423064E + 7
0.43693E + 7 (3.2774)
0.423267E + 7 (4.779E - 2)
0.490321E + 7 (15.8976)
0.433979E + 7 (2.57998)
2 3 4 5
.-. c k-4
H
.-.
ties and mode shapes of the six element model was 2.40 set with the SFEM. Taking the solution of SFEM as the starting value, it took another 2.9 set of iterative perturbation to arrive at the solution of DFEM. On the other hand, the CPU time is 8.50 set for the first five frequencies and mode shapes computed directly from the ten element model by SFEM. Therefore the saving of the CPU time is about 35.0%. Experience shows that the more degrees of freedom of the structure the greater the saving of CPU time.
0
Fig. 2. Cross-section of channel beam.
NUMERICAL
EXAMPLE
Using the formulas and the iterative perturbation presented above a numerical computation have been carried out on a simple supported channel beam as shown in Fig. 2. The physical parameters of this beam are as follows: length L = 300 cm, H = 4.5 cm, B = 2.75 cm, T = 0.5cm. The eigenvalues were calculated and are listed in Table 1; the solutions of the static finite element method (SFEM) were found from eqn (39) and the solutions of the dynamic finite element method (DFEM) were found from eqn (36) and the exact solutions were given by Ref. [l]. From Table 1, it can be seen that the DFEM gives excellent results, and the accuracy of the SFEM is limited in the high frequency range. For example, after four iterations, the error is reduced to 0.53% from the original 10.891% at the fourth mode for the six element model. For the DFEM, the error of the four mode is 0.5336% for the six element model, which is much smaller than that 1.988% of SFEM for the ten element model. This indicated that the DFEM has high accuracy. For this specific example, the accuracy of solutions obtained by DFEM for six elements model is equivalent to those obtained by SFEM for ten elements model. For this example the CPU time required to compute the first five frequen-
CONCLUSIONS
The dynamic finite element method of thin-walled beam with constrained vibration has presented and illustrated by a numerical example of the simple supported channel beam. Based on the numerical results some conclusions can be drawn. Due to the omission of the inertia effect, the eigenvalues obtained by the static finite element method become highly unreliable in the high frequency range. In this range, the dynamic finite element method gives more accurate results. The iterative perturbation proves to be effective and cost saving. REFERENCES I.
2.
3. 4. 5.
6.
J. M. Gere, Torsional vibration of beams of thin-walled open section. J. Appl. Mech. Dec., 334-338 (1954). K. J. Bathe and A. Chaudhary, On the displacement formulation of torsion of shafts with rectangular crosssections. Inf. J. Numer. M&h. 18, 15651573 (1982). R. J. George, Determination of natural frequencies and mode shapes of chasis frames. SAE paper 730504. J. S. Przemieniecki, Theory of Matrix Structural Analysis. McGraw-Hill (1968). Zhang Zongfen and Chen Suhuan, Iterative perturbation for solving nonlinear eigenvalue problem of dynamic reduction. Proceedings of the International Conference on Mechanical Dynamics, 3-6 August 1987. pp. 682-687. W. H. Wittrick and F. W. Williams, A general algorithm for computing natural frequencies of elastic structures. J. Mech. Appl. Math. 24, 263-284 (1971).
& Struchres
in
Vol.39.
No. 6, pi.
597601,
00457949/91 s3.w + 0.00 Pergamon Press pk
1991
GreatBritain.
A NEW METHOD FOR THE VIBRATION THIN-WALLED BEAMS
OF
ZONGFEN ZHANG and &JHUAN CHEN
Department of Mechanics, Jilin University of Technology, Changchun, People’s Republic of China (Reeeiued 19 April
1990)
Ahstraet-This paper presents a new method for thin-walled beams with constrained torsional vibration. Based on the differential equation for torsional vibration, which includes the effect of cross-sectional warping, the shape functions are determined and, in turn, the frequency-dependent mass and stiffness matrices are derived. As an application of the new method, the dynamic finite element method, a numerical example is presented. The results show that this method gives more accurate results in the high-frequency range than those obtained by the static finite-element method.
NOMENCLATURE
Shape function modulus of elasticity indicating element modufus of rigidity (or shear m~ulus) polar moment of inertia of cross-section torsional constant warping constant stiffness matrix mass density mass matrix kinetic energy eigenvector displacement vector velocity vector potential energy angle of twist of beam at any cross-section eigenvalue u 0
Poisson rate natural frequency
INTRODUCTION
Thin-walled beams are wildly used in a broad range of structural applications, such as automobile and aircraft frames. Studies involving static or dynamic calculations of thin-walled beams have been completed by many authors, including Gere [l], Bathe and Chaudhary[2] and George [3]. The common features of these studies is that the static displacement interpolation functions are used for the dynamic analysis. The eigenvalues obtained using the corresponding mass and stiffness matrices are unreliable in the high frequency range. In order to increase
accuracy, we have to abandon the traditional static displacement inte~olation method. From the motion equation of elements, we build the displacement interpolating functions which are the functions of unknown frequency. The corresponding dynamic mass and stiffness matrices are derived. This method
is called the dynamic finite element method. To avoid complicated calculations we assume that the displacement can be expressed as an ascending power series of the unknown frequency [4]. Using this kind of displacement, in the published papers, the dynamic mass and stiffness matrices of bar, beam and plate have been derived. But so far the dynamic mass and stiffness matrices of thin-walled beams with constrained torsional vibration have not been determined. From the differential equation this paper derived the dynamic mass and stiffness matrices of the thinwalled beams with constrained torsional vibration. The application of this method is shown by an example. The results show that the dynamic finite element method has a high accuracy. DIFFERENTIAL EQUATIONS OF THIN-WALLED BEAMS WITH CONSTRAINED TORSIONAL ~BRA~ON
The differential equation for free torsional vibrations of thin-walled open or closed cross-section beams is O(4) -
KQ’2’
.+_ p28’
=
0
,
where K2=-,
E, =P =
GJk 4 J,
p2~!!!% &Jo
E
1 -C2’
1,
P==-J&
for open cross-section, for closed cross section.
(1)
598
Z~NGFEN ZHANGand
SUHUAN CHEN
Assume that
O=tiU,
G
(2)
where (3) and
r\
Lr
u, =e1 x=0
u,=e
u2 =@I x=0
u,=WI x=1
1 x=1
Fig. 1. End displacements of the beam with constrained torsional vibrations.
also Ui, i = 1,2,3,4 are the end displacements of the beam shown in Fig. 1. Substituting eqns (3) and (4) into (2), one gets
=Oda,
(5) 1-u
eqn (5) into (l), we obtain
- pz,io @_d+2qeior= 0. ‘_
(6)
The coefficients of the same powers of w in eqn (6) must be zero, therefore we obtain d&” - K2rjh2)= 0,
(7)
a\” - K2Z\2)= 0,
@a)
1.
@b)
Equations (7) and (8) can be integrated directly. The first matrix rI, is then used to satisfy the boundary conditions that 0 = u, and 8 = u, at x = 0, 6 = uj and 8 =u4 at x = 1, while the remaining matrices - _ must all vanish at x = 0 and x = 1. al,a2,a3,..., The general solution of eqn (7) is as follows: G&c) = ?‘, + F2x + F,Shkx + F,Chkx
F,(l) = A/S,
F2(1) = - KShkl/S,
F,(l) = Shkl,
F,(l) = B/S,
4 (2) = DlKlS,
F,(2) = B/S,
f-3(2)= A IKIS,
F4(2) = -D/K/S,
F,(3) = B/S,
F,(3) = KShkl/S,
F,(3) = -Shkl/S,
F,(3) = -B/S,
F, (4) = HIKIS,
F2 (4)
F,(4) = -B/K/S,
F,(4) = -H/K/S,
(lla)
(11’4
(1lc)
= B/S,
(114
a,,(O) = 1, lX_,=O,
S = KlShKl - 2Chkl -I-2, A = KIShkl - Chkl + 1, B= 1 -Chkl, D = klChk1 - Shkl, H = Shkl - ki. As matrix d, must vanish at x=0 according to eqn (8) we obtain
(9)
requiring that eqn (9) satisfy the boundary conditions yields
UOa)
and x=1,
a, - = 0.
(12)
In order to avoid tedious algebraic manipulation, the following procedure is adopted to solve eqn (8b). To this end, (9) can be expressed as follows: a,,i = F, (i) -I-F2(i)x + F, (i)Shkx + F,(i)Chkx,
i = 1,2, 3,4.
aoz(0) = 09
uo2(l)=0,%
&
i,..=l’~i,_,=o’
(104
where
ni4) - K2fii2’ = p2fio,
aO,(l)=O,% dx /,.,-O,z
=
=I
Substituting eqn (9) into (10) the coefficients in (9) can be found as follows:
Note that for simplicity we define
Substituting
’ dx
=0
(lob)
sub;;~;(ek~_e-kX),2,
Chkx+kX+e-““)/2
(13)
599
Vibration of thin-walled beams Equation (19) can be written as
into eqn (13) gives qi = F, (i) + F2(i)x + (&
+ (F4(i) - F,(i))eekX/2. Substituting leads to a2i=
T(t)‘= ;!?TTM’t?.
(i) + F4(i))ek”/2 (14)
eqn (14) into (8b) and integrating
(8b)
Comparing matrix is
eqn (19) with (20), the element mass
M’=
F,(i)+F,(i)x
+F7(i>x2+F8(i)x3
+ Flo(i)e-k”+ F,,(i)x&
+
‘+ir&dx. I0
F9(i)ek”
+ F,#xewk”,
(15)
(20)
Substituting
(21)
eqn (3) into (21) yields I
M’=
where F,(i) = p2F4(i)/K4 + C,(i),
(16a)
Fe(i) = p2F,(i)/K3/2 + C,(i),
(1W
F,(i) = -p*F, (i)/K2/2,
(16~)
F,(i) = -p2F2(i)/K2/6,
(16d)
F,(i) = -p2(F3(i) + F4(i))/K4/2 + C,(i),
(16e)
F,,(i) = -p2(F4(i) - FI(i))/K4/2 + C,(i),
(160
MZJ$ + f&u2 f . * . y I xo(4+d,w’+...)dx
(22) where A4;=
K,(i) = p2(F4W - 4WY~3/2,
iUS =
=
-P'(F~(z)
-
(1W
Wr))/K3/4,
(17a)
C.,(i) =p2(2 + kl)(F,(i) - F,(i))/K4/4,
(17b)
C,(i) = - C,(i) - C,(i),
(17c)
C,(i) = KC,(i) -kc,(i).
(17d)
J
@I”)2dx/2 + GJk - @I’)*dx/2. (25)
0
Substituting
C,(i) =p2(2 - kl)(F,(i) + F4(i))/K4/4,
(24)
Pl
r1
(16h)
where
’rnZ,(ti~lr,+ Crl$) dx.
I0
From the theory of constrained torsional vibration, the element potential energy is V(tY = E*J,
Fl2G)
(23)
‘mIPa&i&t f0
J0
eqn (2) into (25) we obtain
v(ty = E,J,
’ t7T(cY5iW)t7dx/2 I0 I
+ GJ,
I0
ur(a’%‘)t7dx/2.
(26)
Equation (26) can be written as
Using eqns (1 l)-( 17), 4 and 4 calculated on a digital computer. DYNAMIC
can be easily
Substituting
‘mZ,,~2dx
Comparing eqn (26) with (27) we obtain the element stiffness matrix K’ =
This section presents the calculation of the element mass and stiffness matrices. From the theory of constrained torsional vibration, the element kinetic energy is
s0
(27)
1
MASS AND STIFFNESS MATRICES
T(tY=k
V(t) = ; WK~U.
V-9
’ E,&irTcin
s0
dx $
GJ,li’%’ dX. s0
(28)
Substituting eqn (3) into (28) we get K’ =
’ E, J&i; + o*d; + . . . )r s e,,,.,,;....,,,
eqn (2) into (18) we obtain
s
x (zi;+o%;+***)dx
1
T(t)’ =;
mIPtJTTTdTZitrdx.
0
(1%
=K;+o’K;+...
9
(2%
600
ZONGFENZHANGand
where I K;=
E, J,c?;~&’ dx +
s0
GJ,ci;Tci; dx,
(30)
f0 1
K; =
’ E, J,~i;~ci; dx +
I0
GJkci;T~; dx.
following formulas can be used to calculate MO, Mr, &, and k4 I =
cluster of natural frequencies. In order to avoid the difficulties of cluster frequencies, the iterative perturbation (51 for solving the nonlinear eigenproblem eqn (38) can be used. If AM is dropped, eqn (38) becomes
(31)
J0
The
Mi(i,j)
SUHUAN CHEN
mI, aoiaojdx,
k,u = AM,.
(39)
This is the same eigenproblem as obtained by the static finite element method. Equation (38) can be referred as the perturbation equation of eqn (39). Then the perturbed eigensolutions are
(32)
I0
A; = li + Ali,
(40)
I
M;(i, j) =
mlp (a,, a*, +
azta,)
E, J,a%a&dx
+
dx,
(33)
Au;=ui+Aui,
s 0 I
Ko(i, j) = s 0
GJ,a,&aijdx,
(34)
s 0
(41)
where li and ui are the eigensolutions of eqn (39) Ali and Aui can be easily computed by matrix perturbation
I
K;(i, j) =
E, J,ayia$ dx
-rliu;AMui,
Ali=
s 0
(42)
I
GJka&a$dx,
+
i,j=1,2,3,4.
Aui= i
(35)
The matrices rn; and kg represent the static inertia and stiffness of the thin-walled beam element with constrained torsional vibration, while rni and k; and higher-order terms represent dynamic corrections. ITERATIVE PERTURBATION FOR SOLVING NONLINEAR EIGENPROBLEM
In practical calculations, the iterative perturbation starts with eqn (39), and assume that
For P=l,2,...
THE
AM(p) = w#“(M, - K,) - w\f”(M2 - k.,),
The eigenproblem for thin-walled beams with constrained torsional vibration is as follows:
Al(P)
(36)
(k. + L’K, + . . . )u = l(Mo + AM, + . . . )u,
=
where
fq
A =wr.
=
I
(u!P’)TAM(PMP)v i,s=l,2
i = s,
i(~jp))ThW~hjp),
AajP',
(48)
(37) (49) (3W
or
@ + ‘) = #’ + A@‘,
(50) (51)
k, u = 1 (MO+ AM)u,
AM = 1(M, - M4).
(47)
,..,,
Equation (37) can be rewritten as k,u = l[Mo + 1(M, - K4)]u
(46)
I
a!p + ” = alp) +
=1(Mo+1M2)u.
I?
(45)
Iz.!P’
m
i#s,
-
Neglecting the higher-order terms, we arrive at
_~!P)(U!~))TAM(P)u!P) I I
I -
(&++‘K,)u
(43)
c&u,.
s=,
s 0
(38b)
Equation (38) is a nonlinear eigenproblem. The method for solving eqn (38) has been presented by Wittrick and Williams [6], which was based on the sequence search in every neighbourhood of interested frequencies. However, it is difficult to search a
(52) (53)
where .Cis the convergence tolerance. In general, after four to five iterations, convergence can be achieved.
601
Vibration of thin-walled beams Table 1. The comparison of eigenvalues of simple supported beam with constrained torsional vibration 6 Elements
10 Elements
DFEM (error X)
SFEM (error %)
Modal number
Exact solution
SFEM (error %)
DFEM (error %)
1
O.l37116E+6
0.13727OE + 6 (l.l169E1)
0.137116E + 6 (0)
0.138005E + 6 (0.648356)
O.l37118E+6 (0.14586E - 2)
0.564519E + 6
0.567104E + 6 (4.5786E - 1)
0.564524E + 6 (9.317E - 4)
0.579400E + 6 (2.636)
0.564607E + 6 (0.01588)
0.133037E + 7
0.134458E + 7 (1.0678)
0.133044E + 7 (4.735E - 3)
0.141089E + 7 (6.0592)
0.1331728E + 7 (0.1020766)
0.251495E + 7
0.256495E + 7 (1.988)
0.251537E + 7 (1.661268 - 2)
0.278887E + 7 (10.8917)
0.252837E + 7 (0.5336)
0.423064E + 7
0.43693E + 7 (3.2774)
0.423267E + 7 (4.779E - 2)
0.490321E + 7 (15.8976)
0.433979E + 7 (2.57998)
2 3 4 5
.-. c k-4
H
.-.
ties and mode shapes of the six element model was 2.40 set with the SFEM. Taking the solution of SFEM as the starting value, it took another 2.9 set of iterative perturbation to arrive at the solution of DFEM. On the other hand, the CPU time is 8.50 set for the first five frequencies and mode shapes computed directly from the ten element model by SFEM. Therefore the saving of the CPU time is about 35.0%. Experience shows that the more degrees of freedom of the structure the greater the saving of CPU time.
0
Fig. 2. Cross-section of channel beam.
NUMERICAL
EXAMPLE
Using the formulas and the iterative perturbation presented above a numerical computation have been carried out on a simple supported channel beam as shown in Fig. 2. The physical parameters of this beam are as follows: length L = 300 cm, H = 4.5 cm, B = 2.75 cm, T = 0.5cm. The eigenvalues were calculated and are listed in Table 1; the solutions of the static finite element method (SFEM) were found from eqn (39) and the solutions of the dynamic finite element method (DFEM) were found from eqn (36) and the exact solutions were given by Ref. [l]. From Table 1, it can be seen that the DFEM gives excellent results, and the accuracy of the SFEM is limited in the high frequency range. For example, after four iterations, the error is reduced to 0.53% from the original 10.891% at the fourth mode for the six element model. For the DFEM, the error of the four mode is 0.5336% for the six element model, which is much smaller than that 1.988% of SFEM for the ten element model. This indicated that the DFEM has high accuracy. For this specific example, the accuracy of solutions obtained by DFEM for six elements model is equivalent to those obtained by SFEM for ten elements model. For this example the CPU time required to compute the first five frequen-
CONCLUSIONS
The dynamic finite element method of thin-walled beam with constrained vibration has presented and illustrated by a numerical example of the simple supported channel beam. Based on the numerical results some conclusions can be drawn. Due to the omission of the inertia effect, the eigenvalues obtained by the static finite element method become highly unreliable in the high frequency range. In this range, the dynamic finite element method gives more accurate results. The iterative perturbation proves to be effective and cost saving. REFERENCES I.
2.
3. 4. 5.
6.
J. M. Gere, Torsional vibration of beams of thin-walled open section. J. Appl. Mech. Dec., 334-338 (1954). K. J. Bathe and A. Chaudhary, On the displacement formulation of torsion of shafts with rectangular crosssections. Inf. J. Numer. M&h. 18, 15651573 (1982). R. J. George, Determination of natural frequencies and mode shapes of chasis frames. SAE paper 730504. J. S. Przemieniecki, Theory of Matrix Structural Analysis. McGraw-Hill (1968). Zhang Zongfen and Chen Suhuan, Iterative perturbation for solving nonlinear eigenvalue problem of dynamic reduction. Proceedings of the International Conference on Mechanical Dynamics, 3-6 August 1987. pp. 682-687. W. H. Wittrick and F. W. Williams, A general algorithm for computing natural frequencies of elastic structures. J. Mech. Appl. Math. 24, 263-284 (1971).