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The evaluation of the modal density of paraboloidal and similar shells
Journal of Sound and Vibration (1988) 126(3), 477-483
THE DENSITY
EVALUATION
OF THE
OF PARABOLOIDAL
AND
MODAL SIMILAR
SHELLS
G. H. ELLIOTT Department of Mathematics and Statistics, Portsmouth Polytechnic, Portsmouth PO1 2EG, England (Received 22 December 1987, and in revised form 5 May 1988)
Modal densities of various structures have been found both theoretically in terms of expressions involving integrals, and also experimentally. They are of interest to designers of, for example, satellite aerials and spacecraft structures. The theoretical method is hampered by the difficulty of evaluating the integrals. For certain shells such as a thin plate, spherical cap or circular cylinder, the integrals can be evaluated analytically. In this paper a numerical method is presented for the computation of modal densities for other shell geometries, in particular the paraboloidal shell. Both the isotropic and honeycomb types are considered. Enough details of the numerical method are given to allow the reader to adapt it to other shapes for which analytic integration is not possible.
1. INTRODUCTION The modal density of a structural component is defined as the number of resonant of the modal density is essential in frequencies per bandwidth of 1 Hz. A knowledge assessing the response of the structure to exterior random loads such as rocket motors
in the case of spacecraft components and satellite antennae. Both isotropic and honeycomb shells occur in such structures. Theoretical formulae are known for the modal densities in a number of cases, and are usually expressed as double integrals. These are readily evaluated analytically for simple shell geometries such as the flat plate or spherical cap (see, for example, the papers by Bolotin [l] and Wilkinson [2]). However, antennae frequently approximate more closely to a paraboloidal shell. In the isotropic case, an expression for the modal density can be deduced as a special case of the results of Tulovskii, Aslanian, Kuzina and Lidskii [3]. The formulae are in the form of double integrals but there are difficulties with their evaluation. In this paper numerical methods for computing modal densities of both isotropic and honeycomb paraboloidal shells are presented, and the methods extend to any shape of shell provided its surface formulae are known.
2. THE FORMULAE used in this paper are derived on the basis of the normal deflection of the middle surface of a thin isotropic elastic shell. Tangential inertia forces are not taken into account, and thus the formulae are only involved with the frequencies of transverse vibrations, the longitudinal modes being ignored. Under such assumptions, the model density has been given by Tulovskii, Aslanian, Kuzina and Lidskii [3] as The formulae
477 0022-460X/88/210477+07$03.00/0
@ 1988 Academic Press Limited
478
G.
H. ELLIOTT
where R = (1 - v’)[(sin’ y)/R, + (cos* y)/ R z1’, w = 2~- g is the surface of the shell, (a, p) is a point on the shell where the radii of curvature are R, and R2 and ds* = A* da* + B* dp* (a list of symbols is given in Appendix 3). In certain special cases such as a flat plate, circular cylindrical shell or spherical cap, the integral can be evaluated analytically. However, for the paraboloidal shell, inserting appropriate expressions for RI, R,, A and B gives md _ 4( I - V’)PA Eh loRU I:;2Re{+,2)(;:,,)-,}~rdr’ where the shell has equations x2 + y* = 4~2, p being the focal length and R the semidiameter. Also * .
This integral cannot be evaluated analytically and, except for the higher frequencies, the integral is improper since the denominator vanishes at all points where the integrand changes from real to imaginary.
3. THE NUMERICAL
METHOD
The domain of integration in the (r, y) plane where the integral is real depends on w, and there are three critical frequencies w,, o,, and w,,,. If w > w, the integral is always real and the domain is the whole rectangle (see Figure l(a)). If w, s w < w,, the domain is as shown in Figure l(b) and for o,,, c w s o,, in Figure l(c). For w < wCCC (the cut-off frequency), the integrand is purely imaginary and the modal density is zero. Thus w,,, corresponds to the ring frequency in the case of a cylindrical shell. One can write the integral as
a
laR j:* Re { ~A2-{[cu/(r2+4p2)“2]+[/3/(r2+4p2)3’ dr 2]}2
}(r’+4pl)‘/‘r
dr,
(1)
and A = cow. The where (Y= sin* y, /3 = 4p2 cos’ y, a = 4Jl- u* pao/2pEh denominator of the integrand increases with r and so it remains real if its value at r = 0 is positive: that is, if A2 > 1/2p or w > a/2p, which is w,. The upper limit on y is replaced by yL, where yt retains the value 7r/2 if the integral is real at the position r = R, y = ~12: that is, if o > m/J, which is w,,. Otherwise yL satisfies
(a)
(b)
(c)
Figure 1. Integration domains: (a) for o > w,.; (b) for w,., s w i w, ; (c) for w,,, s w s w,..
MODAL
DENSITY
OF
PARABOLOIDAL
479
SHELLS
The cut-off frequency
occurs when the denominator vanishes at Y = 0, r = R: that is, if w = 0,,4p*/( R* + 4~‘). The transformation Y*+ 4p2 = u* simplifies the integral to o[OyLI,“,Re{,
‘*
]dudy,
(2)
A2-[b/4+W~3)12
where b* = R2+4p2. For fixed 7, the lower limit on u is replaced by u*, where u* =2p ifwaw, and ((Y/U*) + (p/ Use) = A otherwise, where 2p < u* < b. u* is found by Newton’s method, and convergence is guaranteed by the convex nature of the function if the initial value of u is less than u* (say 2~). The singularity at u* is then removed by the transformation ((~/~)+(p/~~)=Asin@
(3)
where 8 = r/2 at u = u* and 6 = 4 (0~ 4 < n/2) at r = b. The transformed n/2 u* de
integral is
I .$ [w~*)+(3PIu4)1’ where for each 8, u( f3) is given implicitly by equation (3), and is again found by Newton’s method. In practise, values of w just greater than w, give slow convergence of the numerical integration procedure since the integrand is almost singular, and so the transformation procedure is also applied if A2 - [(a/u) + (p/u’)] < 0.1, or equivalently, w s w,( 1 + 0*4p*), which is denoted by CW.The integration with respect to y can now proceed. All integrals in the r-direction are finite, and no lack of smoothness in the r-direction has been encountered. The integrals are computed by a Romberg scheme to four significant figures, and with the singularity removed, convergence is fast.
4.
RESULTS
Graphical results for the modal densities of some typical isotropic paraboloidal shells are presented in Figures 2 and 3. Structural details of two shells are given in Appendix 1. Since it is expected that the modal density of a shallow paraboloidal shell might be similar to that of a spherical cap, this quantity is shown for comparison purposes in each case. In the present notation, the formula is md = [Area x J3( 1- ~‘)/h]m/m, where (Y= (l/oR)m, (Y< 1. In making comparisons, a cap with the same radius R has been used and the radius of curvature has been computed give equal surface areas. I
I !O-
o-
\ I I 0
Figure 2. Modal densities:
isotropic
I
500 Frequency
shell type (a). -,
I
1000
(Hz) Paraboloidal
shell; - - -, equivalent
spherical
cap.
480
G. H. ELLIOT
Frequency (Hz)
Figure
3. As Figure 2, but showing
0
Figure 4. Modal densities averaged shell; - - -, equivalent spherical cap.
6
more details
near the peak.
I
I
1000 500 Frequency (Hz)
over intervals
1500
of 100 Hz. Isotropic
shell type (b). -,
Paraboloidal
The main observation to be made from the graphs is that the modal density of a paraboloidal shell is very similar to that of an equivalent spherical cap. The infinite discontinuity at the ring frequency for the cap is replaced by a sharp but finite peak at about the same frequency. For larger frequencies, the modal density tends to that of the spherical cap which in turn is the same as that of the equivalent flat plate. Numerical tests indicate that the peak is lower and less sharp for shells which are less shallow, and are therefore not so spherical. In Figure 4 the results are instead averaged over intervals of 100 Hz, which removes the spherical cap’s discontinuity. The similarity between the paraboloidal and spherical shell is now even greater.
5. HONEYCOMB
PARABOLOIDAL
SHELLS
In this section structures which consist of a core layer covered by thin facing sheets of equal material and thickness are considered. They carry only direct stress, and have no flexural rigidity about their middle surface. However, transverse shear deformation in the core is taken into account via the parameter S which is given below in terms of the structural properties of the shell.
MODAL
DENSITY
OF
PARABOLOIDAL
481
SHELLS
Under these assumptions, a formula for the modal density of a honeycomb shell with constant radii of cervature R, and R2 has been given by Wilkinson [2] as
where j-,+2(1 - YZ)S2
Z -cos2 y+sin2 y -
5(r,y)=Jj:+4(1_v2)s2fi~
fi=R2-hl
R2=4rr2(p,h,+p2h2)h~2/Eh2,
R2
(
R,
>
’
S2= G,G2h:/E,E2h:.
For shallow shells where the radii R, and R2 are in fact slowly varying functions of position on the shell, the results of Tulovskii, Aslanian, Kuzina and Lidskii [3] enable one to extend this formula as in the isotropic case. For the paraboloidal shell for example, the formula becomes j-,+2(1-V2)s2
1 +$)li’
d y)r(
X&+4(1 - V’)Sj-,
dr,
(4)
where y2 is the largest value of y for which the integrand is real. To improve the scaling of the variables, one can write equation (4) as md=K
[oR{y2+~oy2~(r,
y)dy}r(r’+4p’)‘/‘dr,
where K = Z2w/pS,
t(r, y)=[g+2(1--~~)T~]/Jg~+4(1--~~)T~g, .
2
4p2 cos2 y (r2~~pl;‘,2+(r2+4p2)1,2 A2=$(p,:+p2),
T=;,
I
2 7
&-t.
As in the isotopic case, it is necessary to integrate first with respect to r, Changing the order of integration and again substituting u2 = r2+4p2, one has md=K whereb2=R2+4p2,g=A2-[(a/u)+(~/u3)]2, a = sin2 y and p = 4p2 cos2 y. Apart from the change in the integrand, the integration is performed numerically in a very similar manner to that in the isotropic case. The domains of integration have the same form, the critical and cut-off frequencies becoming w, = 112~2, w,, = l/bE and o,,, = (4p*/b*)w,,.
6. RESULTS FOR HONEYCOMB SHELLS Graphical results for the modal densities of a honeycomb paraboloidal shell are presented in Figure 5. Structural details of the shell are given in Appendix 2. As in the isotropic case, the results for an equivalent spherical cap of honeycomb structure are
482
G.
1 0.05 -
H.
ELLIOTT
I
I
I
1
I i
004 r z z 0.031 0.02 -
0.01 -
0
I 1000
I 2000
3000
Frequency (Hz)
Figure 5. Modal densities, honeycomb shell. -,
Paraboloidal shell; - - -, equivalent sphericalcap.
shown for comparison. The graphs show again a similarity between the two shapes, the infinite discontinuity at the ring frequency for the cap being replaced by a finite peak at about the same frequency. For larger frequencies, the modal densities for both shapes tend to that of a flat plate of equal area and construction.
7. CONCLUSION
A method has been proposed which allows the computation of modal densities to be accomplished for any shell for which surface formulae are known as analytic expressions. Only minor modifications are needed to compute the number of resonant modes nf below a given frequency jI These are just the integrated version of the formulae for md, and require the same numerical treatment as the modal density formulae.
REFERENCES 1. V. V. BOLOTIN 1963 Frikladnuya Matematika i Mekhamika 27 (2), 362-364. On the density of the distribution of natural frequencies of thin elastic shells. 2. J. P. D. WILKINSON 1968 Journal of the Acoustical Society of America 43, 245-251. Modal densities of certain shallow structural elements. 3. V. N. TULOVSKII, A. G. ASLANIAN, Z. N. KUZINA and V. B. LIDSKII 1978 Prikkzdnaya Matematika i Mekhamika 37 (4), 604-617. Distribution of the natural frequencies of a thin elastic shell of arbitrary outline.
APPENDIX
1: STRUCTURAL
DETAILS,
ISOTROPIC
SHELL
(a) P = 4.19 m, R = l-6 m (L-Sat C-band Antenna); (b) p = l-12 m, R = O-6 m (Intelsat v West Spot Antenna). In both cases, p = 2720 kg/m3, h = 1 mm, E = 68.9 GN/m3, v = 0.3.
APPENDIX p, = 32-O kg/m3,
h2 = O-24 mm,
GY= 99 MN/m3,
2: STRUCTURAL pt = 1600 kg/m3, E, = 143 GN/m3, V= o-04,
DETAILS,
HONEYCOMB
h, = 11;75 mm, E2 = 133 GN/m3, p=l-12m,
SHELL
G, = 186 MN/m3, R = 0.6 m.
MODAL
DENSITY
APPENDIX
f
E
P
L R, R, 1
h
nf w
Pt Pz
h, 4 md
OF
PARABOLOIDAL
3: LIST
frequency (Hz) Young’s modulus density Poisson ratio shear modulus of core radii of curvature thickness of monocoque element number of resonant modes below f frequency density of core density of facing sheets half-thickness of core thickness of facing sheets modal density
SHELLS
OF SYMBOLS
483
THE DENSITY
EVALUATION
OF THE
OF PARABOLOIDAL
AND
MODAL SIMILAR
SHELLS
G. H. ELLIOTT Department of Mathematics and Statistics, Portsmouth Polytechnic, Portsmouth PO1 2EG, England (Received 22 December 1987, and in revised form 5 May 1988)
Modal densities of various structures have been found both theoretically in terms of expressions involving integrals, and also experimentally. They are of interest to designers of, for example, satellite aerials and spacecraft structures. The theoretical method is hampered by the difficulty of evaluating the integrals. For certain shells such as a thin plate, spherical cap or circular cylinder, the integrals can be evaluated analytically. In this paper a numerical method is presented for the computation of modal densities for other shell geometries, in particular the paraboloidal shell. Both the isotropic and honeycomb types are considered. Enough details of the numerical method are given to allow the reader to adapt it to other shapes for which analytic integration is not possible.
1. INTRODUCTION The modal density of a structural component is defined as the number of resonant of the modal density is essential in frequencies per bandwidth of 1 Hz. A knowledge assessing the response of the structure to exterior random loads such as rocket motors
in the case of spacecraft components and satellite antennae. Both isotropic and honeycomb shells occur in such structures. Theoretical formulae are known for the modal densities in a number of cases, and are usually expressed as double integrals. These are readily evaluated analytically for simple shell geometries such as the flat plate or spherical cap (see, for example, the papers by Bolotin [l] and Wilkinson [2]). However, antennae frequently approximate more closely to a paraboloidal shell. In the isotropic case, an expression for the modal density can be deduced as a special case of the results of Tulovskii, Aslanian, Kuzina and Lidskii [3]. The formulae are in the form of double integrals but there are difficulties with their evaluation. In this paper numerical methods for computing modal densities of both isotropic and honeycomb paraboloidal shells are presented, and the methods extend to any shape of shell provided its surface formulae are known.
2. THE FORMULAE used in this paper are derived on the basis of the normal deflection of the middle surface of a thin isotropic elastic shell. Tangential inertia forces are not taken into account, and thus the formulae are only involved with the frequencies of transverse vibrations, the longitudinal modes being ignored. Under such assumptions, the model density has been given by Tulovskii, Aslanian, Kuzina and Lidskii [3] as The formulae
477 0022-460X/88/210477+07$03.00/0
@ 1988 Academic Press Limited
478
G.
H. ELLIOTT
where R = (1 - v’)[(sin’ y)/R, + (cos* y)/ R z1’, w = 2~- g is the surface of the shell, (a, p) is a point on the shell where the radii of curvature are R, and R2 and ds* = A* da* + B* dp* (a list of symbols is given in Appendix 3). In certain special cases such as a flat plate, circular cylindrical shell or spherical cap, the integral can be evaluated analytically. However, for the paraboloidal shell, inserting appropriate expressions for RI, R,, A and B gives md _ 4( I - V’)PA Eh loRU I:;2Re{+,2)(;:,,)-,}~rdr’ where the shell has equations x2 + y* = 4~2, p being the focal length and R the semidiameter. Also * .
This integral cannot be evaluated analytically and, except for the higher frequencies, the integral is improper since the denominator vanishes at all points where the integrand changes from real to imaginary.
3. THE NUMERICAL
METHOD
The domain of integration in the (r, y) plane where the integral is real depends on w, and there are three critical frequencies w,, o,, and w,,,. If w > w, the integral is always real and the domain is the whole rectangle (see Figure l(a)). If w, s w < w,, the domain is as shown in Figure l(b) and for o,,, c w s o,, in Figure l(c). For w < wCCC (the cut-off frequency), the integrand is purely imaginary and the modal density is zero. Thus w,,, corresponds to the ring frequency in the case of a cylindrical shell. One can write the integral as
a
laR j:* Re { ~A2-{[cu/(r2+4p2)“2]+[/3/(r2+4p2)3’ dr 2]}2
}(r’+4pl)‘/‘r
dr,
(1)
and A = cow. The where (Y= sin* y, /3 = 4p2 cos’ y, a = 4Jl- u* pao/2pEh denominator of the integrand increases with r and so it remains real if its value at r = 0 is positive: that is, if A2 > 1/2p or w > a/2p, which is w,. The upper limit on y is replaced by yL, where yt retains the value 7r/2 if the integral is real at the position r = R, y = ~12: that is, if o > m/J, which is w,,. Otherwise yL satisfies
(a)
(b)
(c)
Figure 1. Integration domains: (a) for o > w,.; (b) for w,., s w i w, ; (c) for w,,, s w s w,..
MODAL
DENSITY
OF
PARABOLOIDAL
479
SHELLS
The cut-off frequency
occurs when the denominator vanishes at Y = 0, r = R: that is, if w = 0,,4p*/( R* + 4~‘). The transformation Y*+ 4p2 = u* simplifies the integral to o[OyLI,“,Re{,
‘*
]dudy,
(2)
A2-[b/4+W~3)12
where b* = R2+4p2. For fixed 7, the lower limit on u is replaced by u*, where u* =2p ifwaw, and ((Y/U*) + (p/ Use) = A otherwise, where 2p < u* < b. u* is found by Newton’s method, and convergence is guaranteed by the convex nature of the function if the initial value of u is less than u* (say 2~). The singularity at u* is then removed by the transformation ((~/~)+(p/~~)=Asin@
(3)
where 8 = r/2 at u = u* and 6 = 4 (0~ 4 < n/2) at r = b. The transformed n/2 u* de
integral is
I .$ [w~*)+(3PIu4)1’ where for each 8, u( f3) is given implicitly by equation (3), and is again found by Newton’s method. In practise, values of w just greater than w, give slow convergence of the numerical integration procedure since the integrand is almost singular, and so the transformation procedure is also applied if A2 - [(a/u) + (p/u’)] < 0.1, or equivalently, w s w,( 1 + 0*4p*), which is denoted by CW.The integration with respect to y can now proceed. All integrals in the r-direction are finite, and no lack of smoothness in the r-direction has been encountered. The integrals are computed by a Romberg scheme to four significant figures, and with the singularity removed, convergence is fast.
4.
RESULTS
Graphical results for the modal densities of some typical isotropic paraboloidal shells are presented in Figures 2 and 3. Structural details of two shells are given in Appendix 1. Since it is expected that the modal density of a shallow paraboloidal shell might be similar to that of a spherical cap, this quantity is shown for comparison purposes in each case. In the present notation, the formula is md = [Area x J3( 1- ~‘)/h]m/m, where (Y= (l/oR)m, (Y< 1. In making comparisons, a cap with the same radius R has been used and the radius of curvature has been computed give equal surface areas. I
I !O-
o-
\ I I 0
Figure 2. Modal densities:
isotropic
I
500 Frequency
shell type (a). -,
I
1000
(Hz) Paraboloidal
shell; - - -, equivalent
spherical
cap.
480
G. H. ELLIOT
Frequency (Hz)
Figure
3. As Figure 2, but showing
0
Figure 4. Modal densities averaged shell; - - -, equivalent spherical cap.
6
more details
near the peak.
I
I
1000 500 Frequency (Hz)
over intervals
1500
of 100 Hz. Isotropic
shell type (b). -,
Paraboloidal
The main observation to be made from the graphs is that the modal density of a paraboloidal shell is very similar to that of an equivalent spherical cap. The infinite discontinuity at the ring frequency for the cap is replaced by a sharp but finite peak at about the same frequency. For larger frequencies, the modal density tends to that of the spherical cap which in turn is the same as that of the equivalent flat plate. Numerical tests indicate that the peak is lower and less sharp for shells which are less shallow, and are therefore not so spherical. In Figure 4 the results are instead averaged over intervals of 100 Hz, which removes the spherical cap’s discontinuity. The similarity between the paraboloidal and spherical shell is now even greater.
5. HONEYCOMB
PARABOLOIDAL
SHELLS
In this section structures which consist of a core layer covered by thin facing sheets of equal material and thickness are considered. They carry only direct stress, and have no flexural rigidity about their middle surface. However, transverse shear deformation in the core is taken into account via the parameter S which is given below in terms of the structural properties of the shell.
MODAL
DENSITY
OF
PARABOLOIDAL
481
SHELLS
Under these assumptions, a formula for the modal density of a honeycomb shell with constant radii of cervature R, and R2 has been given by Wilkinson [2] as
where j-,+2(1 - YZ)S2
Z -cos2 y+sin2 y -
5(r,y)=Jj:+4(1_v2)s2fi~
fi=R2-hl
R2=4rr2(p,h,+p2h2)h~2/Eh2,
R2
(
R,
>
’
S2= G,G2h:/E,E2h:.
For shallow shells where the radii R, and R2 are in fact slowly varying functions of position on the shell, the results of Tulovskii, Aslanian, Kuzina and Lidskii [3] enable one to extend this formula as in the isotropic case. For the paraboloidal shell for example, the formula becomes j-,+2(1-V2)s2
1 +$)li’
d y)r(
X&+4(1 - V’)Sj-,
dr,
(4)
where y2 is the largest value of y for which the integrand is real. To improve the scaling of the variables, one can write equation (4) as md=K
[oR{y2+~oy2~(r,
y)dy}r(r’+4p’)‘/‘dr,
where K = Z2w/pS,
t(r, y)=[g+2(1--~~)T~]/Jg~+4(1--~~)T~g, .
2
4p2 cos2 y (r2~~pl;‘,2+(r2+4p2)1,2 A2=$(p,:+p2),
T=;,
I
2 7
&-t.
As in the isotopic case, it is necessary to integrate first with respect to r, Changing the order of integration and again substituting u2 = r2+4p2, one has md=K whereb2=R2+4p2,g=A2-[(a/u)+(~/u3)]2, a = sin2 y and p = 4p2 cos2 y. Apart from the change in the integrand, the integration is performed numerically in a very similar manner to that in the isotropic case. The domains of integration have the same form, the critical and cut-off frequencies becoming w, = 112~2, w,, = l/bE and o,,, = (4p*/b*)w,,.
6. RESULTS FOR HONEYCOMB SHELLS Graphical results for the modal densities of a honeycomb paraboloidal shell are presented in Figure 5. Structural details of the shell are given in Appendix 2. As in the isotropic case, the results for an equivalent spherical cap of honeycomb structure are
482
G.
1 0.05 -
H.
ELLIOTT
I
I
I
1
I i
004 r z z 0.031 0.02 -
0.01 -
0
I 1000
I 2000
3000
Frequency (Hz)
Figure 5. Modal densities, honeycomb shell. -,
Paraboloidal shell; - - -, equivalent sphericalcap.
shown for comparison. The graphs show again a similarity between the two shapes, the infinite discontinuity at the ring frequency for the cap being replaced by a finite peak at about the same frequency. For larger frequencies, the modal densities for both shapes tend to that of a flat plate of equal area and construction.
7. CONCLUSION
A method has been proposed which allows the computation of modal densities to be accomplished for any shell for which surface formulae are known as analytic expressions. Only minor modifications are needed to compute the number of resonant modes nf below a given frequency jI These are just the integrated version of the formulae for md, and require the same numerical treatment as the modal density formulae.
REFERENCES 1. V. V. BOLOTIN 1963 Frikladnuya Matematika i Mekhamika 27 (2), 362-364. On the density of the distribution of natural frequencies of thin elastic shells. 2. J. P. D. WILKINSON 1968 Journal of the Acoustical Society of America 43, 245-251. Modal densities of certain shallow structural elements. 3. V. N. TULOVSKII, A. G. ASLANIAN, Z. N. KUZINA and V. B. LIDSKII 1978 Prikkzdnaya Matematika i Mekhamika 37 (4), 604-617. Distribution of the natural frequencies of a thin elastic shell of arbitrary outline.
APPENDIX
1: STRUCTURAL
DETAILS,
ISOTROPIC
SHELL
(a) P = 4.19 m, R = l-6 m (L-Sat C-band Antenna); (b) p = l-12 m, R = O-6 m (Intelsat v West Spot Antenna). In both cases, p = 2720 kg/m3, h = 1 mm, E = 68.9 GN/m3, v = 0.3.
APPENDIX p, = 32-O kg/m3,
h2 = O-24 mm,
GY= 99 MN/m3,
2: STRUCTURAL pt = 1600 kg/m3, E, = 143 GN/m3, V= o-04,
DETAILS,
HONEYCOMB
h, = 11;75 mm, E2 = 133 GN/m3, p=l-12m,
SHELL
G, = 186 MN/m3, R = 0.6 m.
MODAL
DENSITY
APPENDIX
f
E
P
L R, R, 1
h
nf w
Pt Pz
h, 4 md
OF
PARABOLOIDAL
3: LIST
frequency (Hz) Young’s modulus density Poisson ratio shear modulus of core radii of curvature thickness of monocoque element number of resonant modes below f frequency density of core density of facing sheets half-thickness of core thickness of facing sheets modal density
SHELLS
OF SYMBOLS
483