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Free axially-symmetric vibrations of thin elastic shells of revolution
FREE
AXIALLY-SYMMETRIC VIBRATIONS OF ELASTIC SHELLS OF REVOLUTION *
THIN
G. I. PSHENICHNOV Moscow (Receiued
15 April
1968)
THE problem is reduced to the search for eigenvalues and eigenfunctions of a set of ordinary linear differential equations with variable coefficients. In papers devoted to the numerical solution of this problem considerable difficulties have been noted, due to the presence of rapidly increasing solutions [ll. In this investigation a numerical method of solving the problem is suggested in which the difficulties indicated have been successfully overcome by using A. A. Abramov’s pivotal condensation method [21. The results of the numerical calculations are compared with solutions based on the asymptotic integration of differential equations [3, 41. 1. For small axially-symmetric have [5l: (BT,)’ .4B(kiT,
harmonic vibrations of shells of revolution we
- B’T2 - ABk,Ni
+ 2Ehk,,zpzABn
+ k2T2) + (BNI)’ + 2Ehko2p2ABw
= 0, -
0,
(1.1)
(BGi)’ - B’G, - ABNi = 0:
here the following notation is introduced in addition to that generally accepted: 2h is the thickness of the shell, R. = 1 /k. some characteristic radius of curvature of the shell, p2 = po2R02I E a dimensionless parameter of frequency of the free vibrations, and p the density of the material of the shell. In (1.1) and further on a prime denotes differentiation with respect to the coordinate a, directed along the meridian of the middle surface of the shell. The elastic relations
have the form 2Eh T2 = l-_((ez
2Eh Ti = 1- _ ~” (Q + a&z),
*Zh.
$chisl.
Mat. mat. Fiz.,
9, 5, 1202-1207,
306
1969.
+ w,
Free
axially-symmetric
vibrations
307
(1.2) 2Ehs Gi =
-
3(1 - 02)
Gz = - 3(;fOZ)
(Xi + c%),
(xz + 0x1),
where the components of the deformation are defined by the formulae u’
w
E, =___--, A
B’u
w
A
E2==-Rz’
Ri
B’Yl
‘yi’
Xi=---,
9
xa=-zr
(1.3)
where (1.4) The set of twelve equations (l.l)-(1.4)
We now introduce the dimensionless
contain twelve unknown functions:
functions (we denote them by letters
with a tilde) by means of the relations G, =
4Eh2G,,
Gz =
N, =
4Eh2&, u =
w =
Rou”,
T, =
2Ehfli,
h =
R&7,
2EhTi,
Tz =
2EhT2,
Y6 =
yi,
R&
In future we shall omit the tilde. We will use the notation YI =
‘St
YZ =
YS =
Ni,
y7 =
Ti,
Gzr
Y& = YS =
u,
Y5 =
w,
Tz.
The boundary conditions for the free vibrations of a shell which contains no vertices can generally be written in the form b’y
=
cay=0
0
for a = l31, for
a=p2,
det (b’b) # 0, det (Cc) # 0,
(1.5)
where
b and c are given numerical matrices and the change to the transposed matrix is denoted by an asterisk.
308
G. I. Pshenichnov
FIG. 1. h/R = 0.025, p2 = 0.7182
FIG. 2. h/R = 0.005, p2 = 0.6348
Thus of the above twelve functions only six enter into the boundary conditions of the problem. Therefore it is advisable to reduce the set of equations ~l.lM1.4~ to 8 set of six different&l equations containing these six unknownfunctions yl, . . . , ye” We obtain such a set of differential equations in the normalform
Free
axially-symmetric
vibrations
-20..
FIG, 3, h/R = 0.001, pz = 0.6076
FIG. 4. h/R = 0.01,pz= 2.273
II II p11***fiS
p=
.
.
.
. .
.
;
psl*.*pss
Paa
=
Pi& =
= PM
PtS =
=
p2i
=
f.tp6 =
PSI
Pa = P&S= PM,’
=:
Pas
=
pa2 = ps$ = p&s= pa* = pss = plc = pss =I 0,
Pii = (f--a)F/fB+h’fh+ (~h)‘/~~, Pi8 = -A / Wtoh, PM = -_%hW= / tUBa,
309
G. I. Pshenichnov
310
pzz = B’lB+ pzr
=
p32 ~34
=
=
~23
B'Ro/BRz,
--Ah,
=
p25
~33
=
I B,
~54
=
ARo
=
=
A(ki
(ko2p2
B+ ~43
psa
Ah,
-p43 =
(I-o)B'/
(Ako2p2 - B’2 I AB2)R0,
=
PI& = pas= d’
(Eh)‘/Eh,
p33
=
+ -
akz), k22),
(Eh)‘/Eh, a2)Ako,
-(1-
Ako,
pai
=
-6 (1 - 02) A / Rob.
Thus the problem consists of finding the eigenvalue pz and eigenfunctions for the differential equation (1.6) taking into account the boundary conditions (1.5).
The values
of the bending
moments
and stresses
in the ring direction
can
be found by the formulae y7 =
2.
oyi + (hB’l6Wye
aya+
(B'RoIWY~--
(RolRz)y3.
solution of the problem under discussion are due to the fact that o -C la(a) < 1. Below, in solving problem (1.5), (1.6) the method used is that described in [21. In this connection we introduce into [PI,
The difficulties
~3 =
@,Itwo
functions
which are solutions
and satisfy
of a numerical
of the equation
the boundary
conditions
cp(b) = b, respectively.
x032) = c.
Then we have the relations
CPU (NY (4 = 0,
X'(U)Y (a) = 0.
The parameter p2 is chosen from the condition that the determinant system (2.1) in y (a) vanishes for some value of a E [IL, Pzl.
(2.1) of the
Also, to determine the eigenfunctions we propose to proceed as follows. By at the points al, ~22, . . . , ~+t numerical integration we obtain cp(c) and x(c) at (aj+f := aj + 6,). Then the relations (2.1) enable us to find 2 = lIzi, . . . . d*
Free
axially-symmetric
3 1I
vibrations
these points, where y(uj) = Cjz (aj), where Cj are as yet unknown numbers. (The function z is determined from five equations of the set (2.1). The remining six equations of this set may serve as checks on the accuracy of calculating z.) By means of equation (1.6) we construct Taylor series at the points aj for some of the functions yi, retaining a reasonable number of first terms, corresponding to Then the comparison the accuracy of the calculation of zi (ai). CjZi(Uj
+
hj/2)
=Cj+*Zi(Uj+i_6j
/2),
j =
1, 2, . . . . m,
(2.2)
enables us to express Cj in terms of one of them. The value of i in (2.2) can vary depending on j: if Cj+i is defined in terms of Cj, then i = k must, for example, be chosen from the condition Izk(ui+i)
1Z IZi(Uj+l) I.
(2.3)
It may happen that the solution so constructed has to be refined, by proceeding from the form of the variability obtained for yi in various parts of the interval of integration (conditions (2.3) for the choice of k must be changed). 3. To determine the frequencies and stress-deformed states in free axiallysymmetric vibrations of shells, whose surfaces are formed by the rotation of second-order of curves around their axes of symmetry, a program based on the above method was written and calculations carried out on a BESM-3M computer. For such shells A=Ri=
RQ
(1+ysin2u)3/2’
B=&sina=
RO sin a
(1 + y sin2 a)%
’
where a is the angle between the normal to the surface of the shell and the axis of revolution, y = const. At a vertex (a = 0) Rt = RZ = Ro. The cases y > --1, y = --1 and y < -1 correspond to ellipses, parabolas and hyperbolas of revolution. The ellipsoid of revolution degenerates into a sphere if y =,o. One of the edges of the shell (CZ= 13~)is free (yi = yz = y3 = o), and the bii = bi2 = other (a = 0,) is rigidly fixed (y, = yS = yJ. In this connection bzS = cl, = cS2 = ce3 = 1; and the remaining coefficients of the matrices b and c are zero. When looking for the eigenfunctions
in (2.3) we took i = 3 or i = 4, because
312
G. I. Pshenichnov
the functions yJ and y, slowly vary [31 for the lowest frequencies of the free vibrations of the shell in the case of regular degeneracy of the system. The first two terms were retained in the expansions (2.2). In the examples of numerical calculations for normalization 01 s
(Y42
+
152)
da
given below we took the condition
=
i,
6
which is useful for seeking solutions of the problem of forced vibrations in the form of an expansion in series of eigenfunctions of the free vibrations of the shell. In one of these examples we investigated the influence of the value of a small parameter h on the solution for fixed values of the other parameters of the problem: v = 0, pi = n / 9, p2 = JC/ 2, (T = 0.3. For free vibrations of the shell with the lowest frequency we obtained p2 = 0.718 (h = 0.025),
p2 = 0.635 (h = 0.005),
p2 = 0.608 (h = O.OOl),
while from asymptotic integration [31 it was found that pZ = 0.587 as h -, 0. The forms of the vibrations (yo, g5) and diagrams of the longitudinal bending moments (y,), corresponding to the values of the small parameter h = 0.025, h = 0.005 and h = 0.001 are given in Figures 1-3, where the dotted lines show the solution by the method described in [3l. If with h = 0.025 this method gives a marked deviation of the required forms of vibrations, with h = 0.001 the solutions obtained by different methods are practically the same (the dotted lines coincide with the continuous lines). In the range of frequencies min k22 < p2 < max k22 the coefficients of the degenerate set of equations (a set of the second order), obtained from (1.6) as h + 0, take on unbounded values of the modulus at points where k22 = p2 (bending points). The graphs of Fig. 4 relate to such a, case for an ellipsoid of revolution with y = 2, #L = a-t16, p2 = 5n / 6, (3 = 0.3, h = 0.01, corresponding to a parameter of frequency of free vibrations p2 = 2.273. Two bending points occur with coordinates & = 0.923 and & = 3t - gl. Translated
by H. F. Cleaves
Free axially-symmetric
313
vibrations
REFERENCES
1.
ZARGAMI, M. S. and ROBINZON, A. R. A numerical method of calculating the free vibrations of spherical shells. Raketnaya tekhn. i kosmonavtika, 5, 7, 51-58, 1967.
2.
ABRAMOV, A. A. On the transfer of boundary conditions for systems of ordinary differential equations (a variant of the pivotal condensation method). Zh. vychisl. Mat. mat. Fiz., 1,3, 542-545, 1961.
3.
PSHENICHNOV, G. I. Small free vibrations of elastic shells of revolution. zh., 5, 4, 685-690, 1965.
4.
PSHENICHNOV, G. I. Free and forced axially-symmetric vibrations of thin elastic shells of revolution. (Tr. VI Vses. konf. po teorii obolochek i plastinokl Part VI of the All-Union conference on the theory of shells and plates. M., Nauka, 707-711, 1967.
5.
GOL’DENVEIZER, A. L. The theory of thin elastic shells. tonkikb obolochek). M., Gostekhizdat, 1953.
Inzh.
(Teoriya uprugikh
AXIALLY-SYMMETRIC VIBRATIONS OF ELASTIC SHELLS OF REVOLUTION *
THIN
G. I. PSHENICHNOV Moscow (Receiued
15 April
1968)
THE problem is reduced to the search for eigenvalues and eigenfunctions of a set of ordinary linear differential equations with variable coefficients. In papers devoted to the numerical solution of this problem considerable difficulties have been noted, due to the presence of rapidly increasing solutions [ll. In this investigation a numerical method of solving the problem is suggested in which the difficulties indicated have been successfully overcome by using A. A. Abramov’s pivotal condensation method [21. The results of the numerical calculations are compared with solutions based on the asymptotic integration of differential equations [3, 41. 1. For small axially-symmetric have [5l: (BT,)’ .4B(kiT,
harmonic vibrations of shells of revolution we
- B’T2 - ABk,Ni
+ 2Ehk,,zpzABn
+ k2T2) + (BNI)’ + 2Ehko2p2ABw
= 0, -
0,
(1.1)
(BGi)’ - B’G, - ABNi = 0:
here the following notation is introduced in addition to that generally accepted: 2h is the thickness of the shell, R. = 1 /k. some characteristic radius of curvature of the shell, p2 = po2R02I E a dimensionless parameter of frequency of the free vibrations, and p the density of the material of the shell. In (1.1) and further on a prime denotes differentiation with respect to the coordinate a, directed along the meridian of the middle surface of the shell. The elastic relations
have the form 2Eh T2 = l-_((ez
2Eh Ti = 1- _ ~” (Q + a&z),
*Zh.
$chisl.
Mat. mat. Fiz.,
9, 5, 1202-1207,
306
1969.
+ w,
Free
axially-symmetric
vibrations
307
(1.2) 2Ehs Gi =
-
3(1 - 02)
Gz = - 3(;fOZ)
(Xi + c%),
(xz + 0x1),
where the components of the deformation are defined by the formulae u’
w
E, =___--, A
B’u
w
A
E2==-Rz’
Ri
B’Yl
‘yi’
Xi=---,
9
xa=-zr
(1.3)
where (1.4) The set of twelve equations (l.l)-(1.4)
We now introduce the dimensionless
contain twelve unknown functions:
functions (we denote them by letters
with a tilde) by means of the relations G, =
4Eh2G,,
Gz =
N, =
4Eh2&, u =
w =
Rou”,
T, =
2Ehfli,
h =
R&7,
2EhTi,
Tz =
2EhT2,
Y6 =
yi,
R&
In future we shall omit the tilde. We will use the notation YI =
‘St
YZ =
YS =
Ni,
y7 =
Ti,
Gzr
Y& = YS =
u,
Y5 =
w,
Tz.
The boundary conditions for the free vibrations of a shell which contains no vertices can generally be written in the form b’y
=
cay=0
0
for a = l31, for
a=p2,
det (b’b) # 0, det (Cc) # 0,
(1.5)
where
b and c are given numerical matrices and the change to the transposed matrix is denoted by an asterisk.
308
G. I. Pshenichnov
FIG. 1. h/R = 0.025, p2 = 0.7182
FIG. 2. h/R = 0.005, p2 = 0.6348
Thus of the above twelve functions only six enter into the boundary conditions of the problem. Therefore it is advisable to reduce the set of equations ~l.lM1.4~ to 8 set of six different&l equations containing these six unknownfunctions yl, . . . , ye” We obtain such a set of differential equations in the normalform
Free
axially-symmetric
vibrations
-20..
FIG, 3, h/R = 0.001, pz = 0.6076
FIG. 4. h/R = 0.01,pz= 2.273
II II p11***fiS
p=
.
.
.
. .
.
;
psl*.*pss
Paa
=
Pi& =
= PM
PtS =
=
p2i
=
f.tp6 =
PSI
Pa = P&S= PM,’
=:
Pas
=
pa2 = ps$ = p&s= pa* = pss = plc = pss =I 0,
Pii = (f--a)F/fB+h’fh+ (~h)‘/~~, Pi8 = -A / Wtoh, PM = -_%hW= / tUBa,
309
G. I. Pshenichnov
310
pzz = B’lB+ pzr
=
p32 ~34
=
=
~23
B'Ro/BRz,
--Ah,
=
p25
~33
=
I B,
~54
=
ARo
=
=
A(ki
(ko2p2
B+ ~43
psa
Ah,
-p43 =
(I-o)B'/
(Ako2p2 - B’2 I AB2)R0,
=
PI& = pas= d’
(Eh)‘/Eh,
p33
=
+ -
akz), k22),
(Eh)‘/Eh, a2)Ako,
-(1-
Ako,
pai
=
-6 (1 - 02) A / Rob.
Thus the problem consists of finding the eigenvalue pz and eigenfunctions for the differential equation (1.6) taking into account the boundary conditions (1.5).
The values
of the bending
moments
and stresses
in the ring direction
can
be found by the formulae y7 =
2.
oyi + (hB’l6Wye
aya+
(B'RoIWY~--
(RolRz)y3.
solution of the problem under discussion are due to the fact that o -C la(a) < 1. Below, in solving problem (1.5), (1.6) the method used is that described in [21. In this connection we introduce into [PI,
The difficulties
~3 =
@,Itwo
functions
which are solutions
and satisfy
of a numerical
of the equation
the boundary
conditions
cp(b) = b, respectively.
x032) = c.
Then we have the relations
CPU (NY (4 = 0,
X'(U)Y (a) = 0.
The parameter p2 is chosen from the condition that the determinant system (2.1) in y (a) vanishes for some value of a E [IL, Pzl.
(2.1) of the
Also, to determine the eigenfunctions we propose to proceed as follows. By at the points al, ~22, . . . , ~+t numerical integration we obtain cp(c) and x(c) at (aj+f := aj + 6,). Then the relations (2.1) enable us to find 2 = lIzi, . . . . d*
Free
axially-symmetric
3 1I
vibrations
these points, where y(uj) = Cjz (aj), where Cj are as yet unknown numbers. (The function z is determined from five equations of the set (2.1). The remining six equations of this set may serve as checks on the accuracy of calculating z.) By means of equation (1.6) we construct Taylor series at the points aj for some of the functions yi, retaining a reasonable number of first terms, corresponding to Then the comparison the accuracy of the calculation of zi (ai). CjZi(Uj
+
hj/2)
=Cj+*Zi(Uj+i_6j
/2),
j =
1, 2, . . . . m,
(2.2)
enables us to express Cj in terms of one of them. The value of i in (2.2) can vary depending on j: if Cj+i is defined in terms of Cj, then i = k must, for example, be chosen from the condition Izk(ui+i)
1Z IZi(Uj+l) I.
(2.3)
It may happen that the solution so constructed has to be refined, by proceeding from the form of the variability obtained for yi in various parts of the interval of integration (conditions (2.3) for the choice of k must be changed). 3. To determine the frequencies and stress-deformed states in free axiallysymmetric vibrations of shells, whose surfaces are formed by the rotation of second-order of curves around their axes of symmetry, a program based on the above method was written and calculations carried out on a BESM-3M computer. For such shells A=Ri=
RQ
(1+ysin2u)3/2’
B=&sina=
RO sin a
(1 + y sin2 a)%
’
where a is the angle between the normal to the surface of the shell and the axis of revolution, y = const. At a vertex (a = 0) Rt = RZ = Ro. The cases y > --1, y = --1 and y < -1 correspond to ellipses, parabolas and hyperbolas of revolution. The ellipsoid of revolution degenerates into a sphere if y =,o. One of the edges of the shell (CZ= 13~)is free (yi = yz = y3 = o), and the bii = bi2 = other (a = 0,) is rigidly fixed (y, = yS = yJ. In this connection bzS = cl, = cS2 = ce3 = 1; and the remaining coefficients of the matrices b and c are zero. When looking for the eigenfunctions
in (2.3) we took i = 3 or i = 4, because
312
G. I. Pshenichnov
the functions yJ and y, slowly vary [31 for the lowest frequencies of the free vibrations of the shell in the case of regular degeneracy of the system. The first two terms were retained in the expansions (2.2). In the examples of numerical calculations for normalization 01 s
(Y42
+
152)
da
given below we took the condition
=
i,
6
which is useful for seeking solutions of the problem of forced vibrations in the form of an expansion in series of eigenfunctions of the free vibrations of the shell. In one of these examples we investigated the influence of the value of a small parameter h on the solution for fixed values of the other parameters of the problem: v = 0, pi = n / 9, p2 = JC/ 2, (T = 0.3. For free vibrations of the shell with the lowest frequency we obtained p2 = 0.718 (h = 0.025),
p2 = 0.635 (h = 0.005),
p2 = 0.608 (h = O.OOl),
while from asymptotic integration [31 it was found that pZ = 0.587 as h -, 0. The forms of the vibrations (yo, g5) and diagrams of the longitudinal bending moments (y,), corresponding to the values of the small parameter h = 0.025, h = 0.005 and h = 0.001 are given in Figures 1-3, where the dotted lines show the solution by the method described in [3l. If with h = 0.025 this method gives a marked deviation of the required forms of vibrations, with h = 0.001 the solutions obtained by different methods are practically the same (the dotted lines coincide with the continuous lines). In the range of frequencies min k22 < p2 < max k22 the coefficients of the degenerate set of equations (a set of the second order), obtained from (1.6) as h + 0, take on unbounded values of the modulus at points where k22 = p2 (bending points). The graphs of Fig. 4 relate to such a, case for an ellipsoid of revolution with y = 2, #L = a-t16, p2 = 5n / 6, (3 = 0.3, h = 0.01, corresponding to a parameter of frequency of free vibrations p2 = 2.273. Two bending points occur with coordinates & = 0.923 and & = 3t - gl. Translated
by H. F. Cleaves
Free axially-symmetric
313
vibrations
REFERENCES
1.
ZARGAMI, M. S. and ROBINZON, A. R. A numerical method of calculating the free vibrations of spherical shells. Raketnaya tekhn. i kosmonavtika, 5, 7, 51-58, 1967.
2.
ABRAMOV, A. A. On the transfer of boundary conditions for systems of ordinary differential equations (a variant of the pivotal condensation method). Zh. vychisl. Mat. mat. Fiz., 1,3, 542-545, 1961.
3.
PSHENICHNOV, G. I. Small free vibrations of elastic shells of revolution. zh., 5, 4, 685-690, 1965.
4.
PSHENICHNOV, G. I. Free and forced axially-symmetric vibrations of thin elastic shells of revolution. (Tr. VI Vses. konf. po teorii obolochek i plastinokl Part VI of the All-Union conference on the theory of shells and plates. M., Nauka, 707-711, 1967.
5.
GOL’DENVEIZER, A. L. The theory of thin elastic shells. tonkikb obolochek). M., Gostekhizdat, 1953.
Inzh.
(Teoriya uprugikh