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Non-linear oscillations of an inextensible, air-inflated, cylindrical membrane
1~. 3. Non-Lmear .Uechanic~, Vol. 23. No. 516. pp. 347-353. 1988 Pnnted m Great Britain.
0
002&74.5?,88 53.00+0.00 1988 Pcrgamon Press plc
NON-LINEAR OSCILLATIONS OF AN INEXTENSIBLE, AIR-INFLATED, CYLINDRICAL MEMBRANE RAYMOND
H. PLAUT and M. JAMES LEEUWRIK
Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (Received 21 October 1987; receiced for publication 15 February 1988) Abstract-A long cylindrical membrane, attached to a horizontal base along two generators and inflated with air, is considered. The material is assumed to be inextensible and its weight is neglected, so that the equilibrium shape of the cross section is circular. Two-dimensional non-linear oscillations about this equilibrium configuration are investigated. Galerkin’s method is applied, first with one term and then with two terms. The method of multiple scales is utilized to study weakly non-linear motions, and the equation of motion is also integrated numerically. The results of these two methods are compared for various cases. It is seen that the vibration frequencies tend to decrease as the amplitude of the motion increases.
1. INTRODUCTION
The non-linear oscillations of a cylindrical membrane are considered. The membrane is anchored along two generators and is inflated by a uniform pressure. Its material is assumed to be inextensible and its weight is neglected, so that the cross-sectional equilibrium shape is circular. The cylinder is assumed to be infinitely long, and two-dimensional oscillations about this circular shape are analyzed (i.e. the vibration mode shapes considered here do not vary along the generators). Linear vibrations of such a cylindrical membrane were analyzed by Firt Cl]. Plaut and Fagan [2] determined the effect of the membrane weight on the linear vibration frequencies and modes. It was found that this effect is a minor one. In the present paper, the non-linear equation of motion, including quadratic and cubic terms, is derived. Galerkin’s method is then applied, first with one term and then with two terms. For weakly non-linear motions, an asymptotic technique, the method of multiple scales, is used, while for larger motions, numerical procedures are utilized. Frequencies are obtained by both methods for increasing response amplitudes, and are presented in tables to demonstrate the influence of the nonlinearities.
2. FORMULATION
Consider a portion of the membrane with unit width along the generators. The crosssectional shape is shown in Fig. 1, where x is the horizontal coordinate, y is the vertical coordinate, s is the arc length from the origin, so is the perimeter, b is the base length, and q is the internal gauge air pressure (force/length).
\ 8 = ill /I
/
/’
/s
,/--
/ ,
so-_
‘\
\
\
q
h(s)
\
\\
\
:
IQ ’ -1
\
\
y
/’
/
I
I
y,
’
+---I
Fig. 1. 347
&x
R. H. PLAUT and M. J. LEEUWRIK
348
The equilibrium configuration is circular, with radius of curvature R, internal angle 8, and central angle Z. The angle between the tangent to the equilibrium shape and the horizontal is denoted $0(s). Note that dl//, -_= ds
-_=A-J.
(1)
In order to analyze motion of the membrane about its equilibrium configuration, consider the element shown in Fig. 2. The tension T, angle $, tangential displacement t’, and normal displacement w are functions of the arc length s and time t. For this portion of the membrane with unit width, the constant mass per length along the cross section is denoted p, The equations of motion in the tangential and normal directions are (2)
I+, = r, and WV,,= W, + 4, s and t denote partial
respectively, where subscripts be written in the form [2]
(3)
derivatives.
The dynamic
II/,=J+(l+u,+Jw)-‘(w,,+J*w)
curvature
can
(4)
where J is defined in (1). Also, the assumption of inextensible relation [2] w=(lj5)-l U,.
material
behavior
leads to the (5)
Equations (2)-(5) can be combined into a single equation for w. First, (3) is solved for T, and the result is differentiated with respect to s and then substituted into (2). Next, the resulting equation is differentiated with respect to s, and (5) is used to eliminate C. Finally, (5) is substituted into (4) and the result is used to eliminate $ in the equation of motion. It is convenient to define the non-dimensional quantities / 5 = ‘4 d(pR ),
W= w/R,
f3=slR,
(6)
where 0 I 0 I a (Fig. 1). If terms in W of fourth order and higher are neglected, of motion for W (0, T) becomes [3]
the equation
W0tJrr- wrt - z&3 - 2 ulg&lz + 2 WBrrZ.9+i wr, (z&3 + 4z) + ztJ* z - 2z,2 + w,,,,zz-2w,,,z,z+
(7)
w,,(2z,2-z,,z-6z2)=o
where z= and subscripts
0 and T represent
partial
w,,+
approximate
solution,
(8)
derivatives.
3. GALERKIN’S
For a one-term
w
ME,THOD
let
w (e, 5) = U (5) sin re,
r = rut/a,
(9)
with n 2 2 to give shapes similar to the linear vibration modes [2]. Equation (9) is substituted into the left side of (7), and the result is multiplied by sin r0, integrated from 6’= 0 to 0 = SI,and set equal to zero. This yields the equation c,ii+c,iiu+c,iiu=+c~u+c~u==o,
(10)
4 w(s(LtL
T(s.t)
A
Fig. 2
T(s+ds,t ‘j’(s+ds.t)
1
Non-linear oscillations of a cylindrical membrane
349
where c, =(?+
l)a/2,
c,=4(r*-l)(r*+2)[1-(-iy]/(3r), c‘$=r*(r*-
CJ= 9(r2 - 1)2x/4,
1)x/2,
(11)
cg = 4r(r2 - I)Z[ t -( - l)“]j3. Note that the quadratic terms vanish (c2 = c5 = 0) for the anti-symmetric functions (n even). For a two-term approximate solution, let W(@,r)=U,(r)sinr,8+U,(r)sinr,t?,
‘j= njnls
(12)
where n, and n, are distinct integers larger than unity. Application of Galerkin’s method leads to the equations
(13)
d,ii,+e,U,+
t
f, (dijUiUj+eijUiUj)
i=* j=l
(14)
where thecoefficients are given in [3]. Ifn, is even, thecoefficients a,,, a,*, b, I, b,,, d12, d21r and ei2+e2, arezero.Ifn~iseven,thecoefficientsa,~,a~,,b,~+b~,,d,,,d~~,e,,,ande~~ are zero. If both n, and n2 are even (i.e. if the two modes are anti-symmetric), the quadratic terms vanish in (13) and (14).
4. ASYMPTOTIC
Weakly non-linear approximation, let
motions
SOLUTION
are considered
in this section. First, for the one-term
U(r) =&U(T)
(15)
in (10). After dividing by EC,, (10) can be written in the form ii-t w*u + shl u* + eh2iiu I- &*h,iiu2 = 0
(16)
where
All quantities in (17) are non-negative, with h, = h,=O Applying the method of multiple scales, let T,=E”?,
h,=c,/c,.
h2=4clr
h, =c&l,
W*=CJCl,
D_=$-,
(17)
if n is even in (9).
m=l,2,.
..,
(18)
m
and u(+uo(To,
T,, . . . )+su,(T,,
T,, . . . )+E~u~(T~,
T,, . . .)+
... .
(19)
Substituting (19) into (16) and setting the coefficients of so, .sl, and .s2 to zero, one .obtains D;uo+o*uo=o, D&i, +w’u, D;u2+w2uz=
= -2DoD~uo-h~u~-h*u~~~uo,
(20) (211
-2D,D,u,-(D;+2DoD2)uo-2h,uou1 -h2uoD;u,--2h2uoDoD,uo-h,u,D;u, -h
3
u2D2u 0 0
(22)
0.
The solution of (20) can be written as u,=Aexp(ioT,)+cc
(23)
R.H. PLAUT and M.J.LEEUWRIK
350
where A=A(T,, T,, . . . ) and cc denotes the complex conjugate of the preceding terms on the right side of the equation. Equation (23) is substituted into (21), and secular terms are eliminated by setting D, A = 0. Thus, one can let A = A( T,) here. The solution of (21) is u,=21A2exp(2ioT,)+2AAA+cc
(24)
where I-=(h, -h2w2)/(6c02),
A=(h,w’-h,)/(2c02),
with I-SO and A 20, and overbars represent complex substituted into (22), and the condition for elimination
(25)
conjugates. Then (23) and (24) are of secular terms is found to be
iD2A+4/.A2A=0
(26)
where i.=[(h,o’-h,)(lOh,-h2w2)-9w’h,]/(24w3).
(27)
Following Nayfeh and Mook ([4], pp. 588.59), one can solve for A. Then, with the use of (15), (18), (19), (23), and (24), one obtains U(r)-Bcos(nr+y)+I-B2COS[2(Rr+y)]
+AB2 (28)
+ Y B3 cos [3(Rr + y)] + O(P) where B and y are constants,
B is of order E,
‘-I’=(4Gh, - 10Gh2m2 -h3d)/(32c02),
(29)
R=w+i.B’.
(30)
and
From (9), (1 l), (17), and (27) one can show that i. < 0 if n 2 2 and 0 < c(< 277,so that this oneterm Galerkin approximation predicts a softening behavior (i.e. the response frequency, R, decreases as the response amplitude, B, increases). Also, for odd values of n, the term AB2 in (28) represents a positive “drift”, i.e. the midpoint of the motion is at a positive value of U rather than at the equilibrium configuration U =O. Next, the method of multiple scales is applied to the two-term approximate solution governed by (13) and (14). Define b, “?‘a,’
rf(rf - 1) (r;+l) ’
where rl, r2 are defined in (12). If the internal the solution is found to be
e, :
o =d,= resonances
r:(r:(r:+l)
1) ’
(31)
w2 z 20, or o1 z 2~~ do not exist,
(32)
(33)
where Bi and yi are constants,
Bi is of order E, and (34) (35)
Non-linear
oscillations
of a cylindrical
membrane
351
The quantities Tij, Aij, ~ij, and Lij are defined in [3]. If ni is even, l-i 1, r12, A1 1, A,,, Q2,, and (I)‘22are zero. If nZ is even, r2 1, Tzz, Azl, AZ2, #ii, and OD,,are zero. If both n, and n2 are even (i.e. the two modes are anti-symmetric), (34) and (35) reduce to Q, =wl-C3w,a,,,l(sa,)lg:,
(36)
Q, = w,-C3wzd,,,l(8d,)lB:.
(37)
Now assume that the two-term approximate solution (12) possesses an internal resonance w1 z 20,, and define a detuning parameter (Tby 02 = 20, -I-EC.
(38)
This occurs, for example, if n, = 2, n, =3, and cz 1.352~. The solvability conditions have the form -2io,D,A,+cc,~,A,exp(ioT,)=O, (39) -2io,D,A,+a,A:exp(-iaT,)=O,
(40)
where ~~=~~:~12+~:~2,--12--2,~/~,~
(41)
~2=M41-ellYdl.
Equations (39) and (40) are the same as (6.2.14) in [4] when damping is absent, and also are a special case of (6.4.18) in [4]. When the solutions are bounded, they can be written in terms of elliptic functions (see [4], pp. 382-385 and 399-402). In general, one can write U, (5) and U2(?), under resonance condition (38), as u,(5)=EV~(T~)cos[ols+e81(T,)]+O(E~),
(42)
v2(5)=EV2(T~)cos[025+e2(T~)]+O(&~),
(43)
where T, = ET and Vi, Bi are governed by the equations dVi ;i~=2
V, V,sinc,
1
1
dV, dT,
-=
Vf sin c,
-$
(45)
2
cm
i,
(46)
with [(T,) defined by 5(T,)=O,-2f?,+aT,.
(47)
If alal < 0, VI and V, may grow with time. Non-trivial steady-state solutions of (44)-(46), if they exist, satisfy sin [ = 0 and Cai WC&)1
-La2 W(4~2)l = kab.
(48)
For the case n1 = 2, n2 = 3, and a = 1.352rr, one finds that o1 = 0.903, w2 = 1.806, ai = 1.825, and a2 = 0.288. One could also consider the internal resonance o2 x 30,. The method of multiple scales leads to solvability conditions having the form (6.3.22) in [4]. However, for any two modes, it turns out that the coefficients of the terms involving the detuning parameter are zero. Hence, the effect of this internal resonance is not manifested at O(E~) and is not very significant.
5.
RESULTS
First, for the one-term Galerkin approximation, (10) was solved numerically for U(r). A Runge-Kutta method was used for central angles a = 7r/2, rc, and 3x/2, and for initial values U(0)=0,0.01, . . . , 0.10 [3]. The steady-state responses were approximately harmonic, with a positive drift for the symmetric modes as described in the previous section.
R. H. PLAUT and M. J. LEEUWRIK
352
Frequencies and amplitudes were computed from the numerical responses. For central angles x = rt and 3n/2, some results are presented in Tables 1 and 2 in columns labeled NUM. In the first row, B is very small and the linear response is obtained. The behavior is softening. For comparison, response frequencies from (30) are also listed in Table 1, in columns labeled MMS. As one would expect, the difference between the results of the asymptotic analysis and the numerical solution tends to increase as the response amplitude grows. Equations (13) and (14), from the two-term Galerkin approximation, were also solved numerically. For small initial values U,(O) and V,(O), the resulting steady-state responses are Table
1. Response
frequencies,
R, in one-term
n=2
approximation;
z= n n=4
n=3
B
NUM
MMS
B
NUM
MMS
B
NUM
MMS
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
1.550 1.548 1.548 1.545 1.541 1.538 1.533 1.527 1.521 1.514 1.505
1.549 1.549 1.547 1.545 1.542 1.537 1.532 1.526 1.519 1.511 1.502
0.000
2.682 2.682 2.679 2.674 2.665 2.651 2.635 2.612 2.590 2.535 2.494
2.683 2.682 2.677 2.673 2.665 2.659 2.650 2.641 2.629 2.614 2.609
O.OQO
3.755 3.747 3.723 3.684 3.629 3.564 3.491 3.403 3.317 3.226 3.135
3.757 3.749 3.724 3.682 3.623 3.548 3.455 3.346 3.220 3.078 2.918
0.009 0.017 0.023 0.030 0.035 0.04 1 0.046 0.052 0.06 1 0.069
Table 2. Response
frequencies,
R, in one-term
n=3
n=2 B
NUM
MMS
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
0.706 0.706 0.706 0.706 0.705 0.705 0.705 0.705 0.704 0.703 0.703
0.706 0.705 0.705 0.705 0.705 0.705 0.705 0.704 0.704 0.703 0.703
-
approximation;
X= 3n/2
n=j
.=4
B
NUM
MMS
0.000 0.010 0.019 0.029 0.038 0.046 0.055 0.063 0.07 1 0.079 0.087
1.549 1.549 1.548 1.548 1.548 1.546 1.545 1.545 1.543 1.541 1.538
1.549 1.549 1.548 1.546 1.543 1.540 1.536 1.532 1.527 1.522 1.516
Table 3. Response
0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
frequencies,
-
B
NUM
MMS
B
NUM
MMS
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
2.315 2.315 2.307 2.300 2.285 2.273 2.254 2.229 2.205 2.183 2.155
2.315 2.313 2.308 2.299 2.286 2.270 2.250 2.227 2.200 2.169 2.135
0.000 0.009 0.017 0.024 0.030 0.036 0.042 0.047 0.052 0.057 0.062
3.046 3.039 3.027 3.008 2.989 2.963 2.932 2.904 2.868 2.832 2.790
3.046 3.043 3.034 3.022 3.008 2.991 2.97 I 2.953 2.932 2.909 2.883
52, and RL, in two-term
approximation;
0, “1
n2
B,
B*
2 2 2 2 2 2 2 2 2 3 3 3
3 3 3
0.000
0.000
0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 O.C00 0.005 0.010 0.000 0.005 0.010
0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 O.OOG 0.005 0.010 O.OQO 0.005 0.010
3
3 3 4 4 4
4
4 4 5 5 5 4 4 4
5 5 5 5 5 5
z= n
Q*
NUM
MMS
NUM
MMS
1.550 1.548 1.541 1.550 1.536 1.496 1.550 1.503 1.285 2.682 2.677 2.662 2.682 2.668 2.607 3.755 3.743 3.708
1.549 1.549 1.546 1.549 1.549 1.549 1.549 1.557 1.579 2.683 2.679 2.665 2.683 2.705 2.769 3.757 3.752 3.737
2.682 2.682 2.679 3.755 3.755 3.747 4.808 4.803 4.788 3.755 3.755 3.747 4.808 4.798 4.784 4.808 4.798 4.777
2.683 2.682 2.677 3.757 3.755 3.749 4.804 4.793 4.762 3.757 3.754 3.744 4.804 4.793 4.759 4.804 4.794 4.762
Non-linear Table 4. Response
“1
n2
2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4
3 3 3 4 4 4 5 5 5 4 4 4 5 5 5 5 5 5
oscillations
frequencies,
S, 0.000 0.005 0.010 0.000 0.005 0.010 o.cKKl 0.005 0.010 O.COO 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010
of a cylindrical
353
membrane
R, and R,, in two-term
approximation;
2 = 3n/2
BZ
NUM
MMS
NUM
MMS
0.000 0.005 0.010 O.ooO 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 O.ooO 0.005 0.010
0.706 0.706 0.706 0.706 0.703 0.694 0.706 0.694 0.661 1.550 1.548 1.545 1.550 1.543 1.524 2.315 2.312 2.305
0.706 0.706 0.706 0.706 0.706 0.706 0.706 0.706 0.708 1.549 1.549 1.548 1.549 1.551 1.556 2.315 2.314 2.312
1.550 I .550 1.548 2.315 2.315 2.315 3.046 3.046 3.042 2.315 2.315 2.312 3.046 3.046 3.039 3.046 3.046 3.039
1.549 1.549 1.548 2.315 2.314 2.313 3.046 3.044 3.039 2.315 .2.3 14 2.312 3.046 3.044 3.038 3.046 3.044 3.039
approximately harmonic. Again, the behavior is softening. Central angles a = n/2, 71, and 3x/2 were considered [3]. Some numerical integration results for a= n and 3n/2 are presented in Tables 3 and 4 in columns labeled NUM. Frequencies R, and Q, from the asymptotic analysis, given by (34) and (3.9, are listed in columns labeled MMS. They also indicate softening behavior except for the cases rot =2, n, =.5 and n, =3, n2 =5, when R, increases with increasing amplitude. Ackno~vlecigemenr-This No. MSM-8505853.
research
was
supported
by the
U.S.
National
Science
Foundation
under
Grant
REFERENCES V. Firt, Sratics, Formfinding and Dynamics ofAir-Supported Membrane Structures. Martinus NijhoK The Hague (1983). R. H. Plaut and T. D. Fagan, Vibrations ofan inextensible, air-inflated, cylindrical membrane. J. appl. ,\fech.. in press. M. J. Leeuwrik, Nonlinear vibration analysis of inflatable dams. M.S. Thesis, Virginia Polytechnic Institute and State University, Blacksburg. Virginia (1987). A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations. Wiley-Interscience, New York (1979).
0
002&74.5?,88 53.00+0.00 1988 Pcrgamon Press plc
NON-LINEAR OSCILLATIONS OF AN INEXTENSIBLE, AIR-INFLATED, CYLINDRICAL MEMBRANE RAYMOND
H. PLAUT and M. JAMES LEEUWRIK
Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (Received 21 October 1987; receiced for publication 15 February 1988) Abstract-A long cylindrical membrane, attached to a horizontal base along two generators and inflated with air, is considered. The material is assumed to be inextensible and its weight is neglected, so that the equilibrium shape of the cross section is circular. Two-dimensional non-linear oscillations about this equilibrium configuration are investigated. Galerkin’s method is applied, first with one term and then with two terms. The method of multiple scales is utilized to study weakly non-linear motions, and the equation of motion is also integrated numerically. The results of these two methods are compared for various cases. It is seen that the vibration frequencies tend to decrease as the amplitude of the motion increases.
1. INTRODUCTION
The non-linear oscillations of a cylindrical membrane are considered. The membrane is anchored along two generators and is inflated by a uniform pressure. Its material is assumed to be inextensible and its weight is neglected, so that the cross-sectional equilibrium shape is circular. The cylinder is assumed to be infinitely long, and two-dimensional oscillations about this circular shape are analyzed (i.e. the vibration mode shapes considered here do not vary along the generators). Linear vibrations of such a cylindrical membrane were analyzed by Firt Cl]. Plaut and Fagan [2] determined the effect of the membrane weight on the linear vibration frequencies and modes. It was found that this effect is a minor one. In the present paper, the non-linear equation of motion, including quadratic and cubic terms, is derived. Galerkin’s method is then applied, first with one term and then with two terms. For weakly non-linear motions, an asymptotic technique, the method of multiple scales, is used, while for larger motions, numerical procedures are utilized. Frequencies are obtained by both methods for increasing response amplitudes, and are presented in tables to demonstrate the influence of the nonlinearities.
2. FORMULATION
Consider a portion of the membrane with unit width along the generators. The crosssectional shape is shown in Fig. 1, where x is the horizontal coordinate, y is the vertical coordinate, s is the arc length from the origin, so is the perimeter, b is the base length, and q is the internal gauge air pressure (force/length).
\ 8 = ill /I
/
/’
/s
,/--
/ ,
so-_
‘\
\
\
q
h(s)
\
\\
\
:
IQ ’ -1
\
\
y
/’
/
I
I
y,
’
+---I
Fig. 1. 347
&x
R. H. PLAUT and M. J. LEEUWRIK
348
The equilibrium configuration is circular, with radius of curvature R, internal angle 8, and central angle Z. The angle between the tangent to the equilibrium shape and the horizontal is denoted $0(s). Note that dl//, -_= ds
-_=A-J.
(1)
In order to analyze motion of the membrane about its equilibrium configuration, consider the element shown in Fig. 2. The tension T, angle $, tangential displacement t’, and normal displacement w are functions of the arc length s and time t. For this portion of the membrane with unit width, the constant mass per length along the cross section is denoted p, The equations of motion in the tangential and normal directions are (2)
I+, = r, and WV,,= W, + 4, s and t denote partial
respectively, where subscripts be written in the form [2]
(3)
derivatives.
The dynamic
II/,=J+(l+u,+Jw)-‘(w,,+J*w)
curvature
can
(4)
where J is defined in (1). Also, the assumption of inextensible relation [2] w=(lj5)-l U,.
material
behavior
leads to the (5)
Equations (2)-(5) can be combined into a single equation for w. First, (3) is solved for T, and the result is differentiated with respect to s and then substituted into (2). Next, the resulting equation is differentiated with respect to s, and (5) is used to eliminate C. Finally, (5) is substituted into (4) and the result is used to eliminate $ in the equation of motion. It is convenient to define the non-dimensional quantities / 5 = ‘4 d(pR ),
W= w/R,
f3=slR,
(6)
where 0 I 0 I a (Fig. 1). If terms in W of fourth order and higher are neglected, of motion for W (0, T) becomes [3]
the equation
W0tJrr- wrt - z&3 - 2 ulg&lz + 2 WBrrZ.9+i wr, (z&3 + 4z) + ztJ* z - 2z,2 + w,,,,zz-2w,,,z,z+
(7)
w,,(2z,2-z,,z-6z2)=o
where z= and subscripts
0 and T represent
partial
w,,+
approximate
solution,
(8)
derivatives.
3. GALERKIN’S
For a one-term
w
ME,THOD
let
w (e, 5) = U (5) sin re,
r = rut/a,
(9)
with n 2 2 to give shapes similar to the linear vibration modes [2]. Equation (9) is substituted into the left side of (7), and the result is multiplied by sin r0, integrated from 6’= 0 to 0 = SI,and set equal to zero. This yields the equation c,ii+c,iiu+c,iiu=+c~u+c~u==o,
(10)
4 w(s(LtL
T(s.t)
A
Fig. 2
T(s+ds,t ‘j’(s+ds.t)
1
Non-linear oscillations of a cylindrical membrane
349
where c, =(?+
l)a/2,
c,=4(r*-l)(r*+2)[1-(-iy]/(3r), c‘$=r*(r*-
CJ= 9(r2 - 1)2x/4,
1)x/2,
(11)
cg = 4r(r2 - I)Z[ t -( - l)“]j3. Note that the quadratic terms vanish (c2 = c5 = 0) for the anti-symmetric functions (n even). For a two-term approximate solution, let W(@,r)=U,(r)sinr,8+U,(r)sinr,t?,
‘j= njnls
(12)
where n, and n, are distinct integers larger than unity. Application of Galerkin’s method leads to the equations
(13)
d,ii,+e,U,+
t
f, (dijUiUj+eijUiUj)
i=* j=l
(14)
where thecoefficients are given in [3]. Ifn, is even, thecoefficients a,,, a,*, b, I, b,,, d12, d21r and ei2+e2, arezero.Ifn~iseven,thecoefficientsa,~,a~,,b,~+b~,,d,,,d~~,e,,,ande~~ are zero. If both n, and n2 are even (i.e. if the two modes are anti-symmetric), the quadratic terms vanish in (13) and (14).
4. ASYMPTOTIC
Weakly non-linear approximation, let
motions
SOLUTION
are considered
in this section. First, for the one-term
U(r) =&U(T)
(15)
in (10). After dividing by EC,, (10) can be written in the form ii-t w*u + shl u* + eh2iiu I- &*h,iiu2 = 0
(16)
where
All quantities in (17) are non-negative, with h, = h,=O Applying the method of multiple scales, let T,=E”?,
h,=c,/c,.
h2=4clr
h, =c&l,
W*=CJCl,
D_=$-,
(17)
if n is even in (9).
m=l,2,.
..,
(18)
m
and u(+uo(To,
T,, . . . )+su,(T,,
T,, . . . )+E~u~(T~,
T,, . . .)+
... .
(19)
Substituting (19) into (16) and setting the coefficients of so, .sl, and .s2 to zero, one .obtains D;uo+o*uo=o, D&i, +w’u, D;u2+w2uz=
= -2DoD~uo-h~u~-h*u~~~uo,
(20) (211
-2D,D,u,-(D;+2DoD2)uo-2h,uou1 -h2uoD;u,--2h2uoDoD,uo-h,u,D;u, -h
3
u2D2u 0 0
(22)
0.
The solution of (20) can be written as u,=Aexp(ioT,)+cc
(23)
R.H. PLAUT and M.J.LEEUWRIK
350
where A=A(T,, T,, . . . ) and cc denotes the complex conjugate of the preceding terms on the right side of the equation. Equation (23) is substituted into (21), and secular terms are eliminated by setting D, A = 0. Thus, one can let A = A( T,) here. The solution of (21) is u,=21A2exp(2ioT,)+2AAA+cc
(24)
where I-=(h, -h2w2)/(6c02),
A=(h,w’-h,)/(2c02),
with I-SO and A 20, and overbars represent complex substituted into (22), and the condition for elimination
(25)
conjugates. Then (23) and (24) are of secular terms is found to be
iD2A+4/.A2A=0
(26)
where i.=[(h,o’-h,)(lOh,-h2w2)-9w’h,]/(24w3).
(27)
Following Nayfeh and Mook ([4], pp. 588.59), one can solve for A. Then, with the use of (15), (18), (19), (23), and (24), one obtains U(r)-Bcos(nr+y)+I-B2COS[2(Rr+y)]
+AB2 (28)
+ Y B3 cos [3(Rr + y)] + O(P) where B and y are constants,
B is of order E,
‘-I’=(4Gh, - 10Gh2m2 -h3d)/(32c02),
(29)
R=w+i.B’.
(30)
and
From (9), (1 l), (17), and (27) one can show that i. < 0 if n 2 2 and 0 < c(< 277,so that this oneterm Galerkin approximation predicts a softening behavior (i.e. the response frequency, R, decreases as the response amplitude, B, increases). Also, for odd values of n, the term AB2 in (28) represents a positive “drift”, i.e. the midpoint of the motion is at a positive value of U rather than at the equilibrium configuration U =O. Next, the method of multiple scales is applied to the two-term approximate solution governed by (13) and (14). Define b, “?‘a,’
rf(rf - 1) (r;+l) ’
where rl, r2 are defined in (12). If the internal the solution is found to be
e, :
o =d,= resonances
r:(r:(r:+l)
1) ’
(31)
w2 z 20, or o1 z 2~~ do not exist,
(32)
(33)
where Bi and yi are constants,
Bi is of order E, and (34) (35)
Non-linear
oscillations
of a cylindrical
membrane
351
The quantities Tij, Aij, ~ij, and Lij are defined in [3]. If ni is even, l-i 1, r12, A1 1, A,,, Q2,, and (I)‘22are zero. If nZ is even, r2 1, Tzz, Azl, AZ2, #ii, and OD,,are zero. If both n, and n2 are even (i.e. the two modes are anti-symmetric), (34) and (35) reduce to Q, =wl-C3w,a,,,l(sa,)lg:,
(36)
Q, = w,-C3wzd,,,l(8d,)lB:.
(37)
Now assume that the two-term approximate solution (12) possesses an internal resonance w1 z 20,, and define a detuning parameter (Tby 02 = 20, -I-EC.
(38)
This occurs, for example, if n, = 2, n, =3, and cz 1.352~. The solvability conditions have the form -2io,D,A,+cc,~,A,exp(ioT,)=O, (39) -2io,D,A,+a,A:exp(-iaT,)=O,
(40)
where ~~=~~:~12+~:~2,--12--2,~/~,~
(41)
~2=M41-ellYdl.
Equations (39) and (40) are the same as (6.2.14) in [4] when damping is absent, and also are a special case of (6.4.18) in [4]. When the solutions are bounded, they can be written in terms of elliptic functions (see [4], pp. 382-385 and 399-402). In general, one can write U, (5) and U2(?), under resonance condition (38), as u,(5)=EV~(T~)cos[ols+e81(T,)]+O(E~),
(42)
v2(5)=EV2(T~)cos[025+e2(T~)]+O(&~),
(43)
where T, = ET and Vi, Bi are governed by the equations dVi ;i~=2
V, V,sinc,
1
1
dV, dT,
-=
Vf sin c,
-$
(45)
2
cm
i,
(46)
with [(T,) defined by 5(T,)=O,-2f?,+aT,.
(47)
If alal < 0, VI and V, may grow with time. Non-trivial steady-state solutions of (44)-(46), if they exist, satisfy sin [ = 0 and Cai WC&)1
-La2 W(4~2)l = kab.
(48)
For the case n1 = 2, n2 = 3, and a = 1.352rr, one finds that o1 = 0.903, w2 = 1.806, ai = 1.825, and a2 = 0.288. One could also consider the internal resonance o2 x 30,. The method of multiple scales leads to solvability conditions having the form (6.3.22) in [4]. However, for any two modes, it turns out that the coefficients of the terms involving the detuning parameter are zero. Hence, the effect of this internal resonance is not manifested at O(E~) and is not very significant.
5.
RESULTS
First, for the one-term Galerkin approximation, (10) was solved numerically for U(r). A Runge-Kutta method was used for central angles a = 7r/2, rc, and 3x/2, and for initial values U(0)=0,0.01, . . . , 0.10 [3]. The steady-state responses were approximately harmonic, with a positive drift for the symmetric modes as described in the previous section.
R. H. PLAUT and M. J. LEEUWRIK
352
Frequencies and amplitudes were computed from the numerical responses. For central angles x = rt and 3n/2, some results are presented in Tables 1 and 2 in columns labeled NUM. In the first row, B is very small and the linear response is obtained. The behavior is softening. For comparison, response frequencies from (30) are also listed in Table 1, in columns labeled MMS. As one would expect, the difference between the results of the asymptotic analysis and the numerical solution tends to increase as the response amplitude grows. Equations (13) and (14), from the two-term Galerkin approximation, were also solved numerically. For small initial values U,(O) and V,(O), the resulting steady-state responses are Table
1. Response
frequencies,
R, in one-term
n=2
approximation;
z= n n=4
n=3
B
NUM
MMS
B
NUM
MMS
B
NUM
MMS
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
1.550 1.548 1.548 1.545 1.541 1.538 1.533 1.527 1.521 1.514 1.505
1.549 1.549 1.547 1.545 1.542 1.537 1.532 1.526 1.519 1.511 1.502
0.000
2.682 2.682 2.679 2.674 2.665 2.651 2.635 2.612 2.590 2.535 2.494
2.683 2.682 2.677 2.673 2.665 2.659 2.650 2.641 2.629 2.614 2.609
O.OQO
3.755 3.747 3.723 3.684 3.629 3.564 3.491 3.403 3.317 3.226 3.135
3.757 3.749 3.724 3.682 3.623 3.548 3.455 3.346 3.220 3.078 2.918
0.009 0.017 0.023 0.030 0.035 0.04 1 0.046 0.052 0.06 1 0.069
Table 2. Response
frequencies,
R, in one-term
n=3
n=2 B
NUM
MMS
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
0.706 0.706 0.706 0.706 0.705 0.705 0.705 0.705 0.704 0.703 0.703
0.706 0.705 0.705 0.705 0.705 0.705 0.705 0.704 0.704 0.703 0.703
-
approximation;
X= 3n/2
n=j
.=4
B
NUM
MMS
0.000 0.010 0.019 0.029 0.038 0.046 0.055 0.063 0.07 1 0.079 0.087
1.549 1.549 1.548 1.548 1.548 1.546 1.545 1.545 1.543 1.541 1.538
1.549 1.549 1.548 1.546 1.543 1.540 1.536 1.532 1.527 1.522 1.516
Table 3. Response
0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
frequencies,
-
B
NUM
MMS
B
NUM
MMS
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
2.315 2.315 2.307 2.300 2.285 2.273 2.254 2.229 2.205 2.183 2.155
2.315 2.313 2.308 2.299 2.286 2.270 2.250 2.227 2.200 2.169 2.135
0.000 0.009 0.017 0.024 0.030 0.036 0.042 0.047 0.052 0.057 0.062
3.046 3.039 3.027 3.008 2.989 2.963 2.932 2.904 2.868 2.832 2.790
3.046 3.043 3.034 3.022 3.008 2.991 2.97 I 2.953 2.932 2.909 2.883
52, and RL, in two-term
approximation;
0, “1
n2
B,
B*
2 2 2 2 2 2 2 2 2 3 3 3
3 3 3
0.000
0.000
0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 O.C00 0.005 0.010 0.000 0.005 0.010
0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 O.OOG 0.005 0.010 O.OQO 0.005 0.010
3
3 3 4 4 4
4
4 4 5 5 5 4 4 4
5 5 5 5 5 5
z= n
Q*
NUM
MMS
NUM
MMS
1.550 1.548 1.541 1.550 1.536 1.496 1.550 1.503 1.285 2.682 2.677 2.662 2.682 2.668 2.607 3.755 3.743 3.708
1.549 1.549 1.546 1.549 1.549 1.549 1.549 1.557 1.579 2.683 2.679 2.665 2.683 2.705 2.769 3.757 3.752 3.737
2.682 2.682 2.679 3.755 3.755 3.747 4.808 4.803 4.788 3.755 3.755 3.747 4.808 4.798 4.784 4.808 4.798 4.777
2.683 2.682 2.677 3.757 3.755 3.749 4.804 4.793 4.762 3.757 3.754 3.744 4.804 4.793 4.759 4.804 4.794 4.762
Non-linear Table 4. Response
“1
n2
2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4
3 3 3 4 4 4 5 5 5 4 4 4 5 5 5 5 5 5
oscillations
frequencies,
S, 0.000 0.005 0.010 0.000 0.005 0.010 o.cKKl 0.005 0.010 O.COO 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010
of a cylindrical
353
membrane
R, and R,, in two-term
approximation;
2 = 3n/2
BZ
NUM
MMS
NUM
MMS
0.000 0.005 0.010 O.ooO 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 0.000 0.005 0.010 O.ooO 0.005 0.010
0.706 0.706 0.706 0.706 0.703 0.694 0.706 0.694 0.661 1.550 1.548 1.545 1.550 1.543 1.524 2.315 2.312 2.305
0.706 0.706 0.706 0.706 0.706 0.706 0.706 0.706 0.708 1.549 1.549 1.548 1.549 1.551 1.556 2.315 2.314 2.312
1.550 I .550 1.548 2.315 2.315 2.315 3.046 3.046 3.042 2.315 2.315 2.312 3.046 3.046 3.039 3.046 3.046 3.039
1.549 1.549 1.548 2.315 2.314 2.313 3.046 3.044 3.039 2.315 .2.3 14 2.312 3.046 3.044 3.038 3.046 3.044 3.039
approximately harmonic. Again, the behavior is softening. Central angles a = n/2, 71, and 3x/2 were considered [3]. Some numerical integration results for a= n and 3n/2 are presented in Tables 3 and 4 in columns labeled NUM. Frequencies R, and Q, from the asymptotic analysis, given by (34) and (3.9, are listed in columns labeled MMS. They also indicate softening behavior except for the cases rot =2, n, =.5 and n, =3, n2 =5, when R, increases with increasing amplitude. Ackno~vlecigemenr-This No. MSM-8505853.
research
was
supported
by the
U.S.
National
Science
Foundation
under
Grant
REFERENCES V. Firt, Sratics, Formfinding and Dynamics ofAir-Supported Membrane Structures. Martinus NijhoK The Hague (1983). R. H. Plaut and T. D. Fagan, Vibrations ofan inextensible, air-inflated, cylindrical membrane. J. appl. ,\fech.. in press. M. J. Leeuwrik, Nonlinear vibration analysis of inflatable dams. M.S. Thesis, Virginia Polytechnic Institute and State University, Blacksburg. Virginia (1987). A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations. Wiley-Interscience, New York (1979).