Design and Stress Analysis of Extraterrestrial ... - The Black Vault

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The peculiarity. of Green's function is its substance and
continuity along the beam,
i.e. , , over the range
Furthermore, lfor any elastic system, Green's functions hdve
the property of symmetry or reciprocity,, i.e., K(z, s) = K(s, z).
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*In the theory of integral equatAons it is proven that:
a homogeneous integral equationhas an Infinite, set of eigenvalues
ifor X;
- all the eigenvalues of X are real. The physical sense of X
is
the square of the natural frequencies pf a system:
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a
(dn2
- each'eigenvalue df A corresponds to at least one eigenfunction;
the number of linearly independent.'eigenfunctions which correspond
to.a given eigenvalue is'finite;,
- the eigenfunctions of, an integral equation With a symmetrical
nucleu$ form'complete normal. and orthogonal (orthc-ormalized) sets
uf fundtions bound by the relation
a
I
Sa.iyjdz
a " pj i
(3 .39)
npH =
when
where q, is
the generalized transverse load..
The obtained linear integral equation (3.37) has the following
properties: I
- it has an infinite numbex'of eigenvalues c2 cl 1' W2 022 w •c3 c: 2
''
i.e., the 'ibrating glade has an infinite number of natural frequencies;
- eaph eigenvalue corresponds ýo a strictly defined eigenfunction
Ul, 'u2, U 3 ... , ,i.e., its own form of elastic line;
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