Page 574 - Aircraft Stuctures for Engineering Student

13.3 Structural vibration 555

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Fig. 13.9 Massheam system for Example 13.1.

where wl and v2 are the vertical displacements of the masses at any instant of time.
In this example, displacements are-assumed to be caused by bending strains only;
the flexibility coefficients Sll, S2, and S12(= 621) may therefore be found by the
unit load method described in Section 4.8. From the first of Eqs (4.27) we deduce
that

(iii)

where Mi is the bending moment at any section z due to a unit load at the point i and
Mi is the bending moment at any section z produced by a unit load at the point j.
Therefore, from Fig. 13.9
M1 = l(1-z) O<Z<l
M2 = 1(1/2 - z) 0 < z < 112

M2=0 112 < z < 1
Hence

Integrating Eqs (iv), (v) and (vi) and substituting limits, we obtain

13 i3 513
611 =E 622 =- s -6 -
l2 - 21 - 48EI
24EI ’
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Each mass describes simple harmonic motion in the normal modes of oscillation so
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that w1 = v’: sin(wt + E) and v2 = v! sin(wt + E). Hence iil = -w w1 and 32 = -&2.
Substituting for ij,, w2, Sll, S2, and S12(= in Eqs (i) and (ii) and writing
X = nzI3/(3 x 48EI), we obtain
(1 - 16Xu2)vl - 15Xw2v1 = 0 (4
5XW2Vl - (1 - ~XW’)W~ = 0 (viii)