Page 410 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.7 Coupled Flexure-Torsion Vibration 397

To start the computation, we note that the moment and shear at station 1 are
zero. We can choose the deflection at station 1 to be 1.0, in which case the slope at

this point becomes an unknown 6. We, therefore, carry out two columns of calcula­

tions for each quantity, starting with y, = 1.0, 0, == 0, and yj = 0, 6^ = 6. The
unknown slope 6^ = 6 is found by forcing 0^ at the fixed end to be zero, after which
the deflection can be calculated and plotted against co. The natural frequencies of
the system are those for which = 0.
To search for the natural frequencies, computer calculations were made be­
tween o) = 10 to io = 400 at frequency steps of 10 rad/s. Tabulation of y^ versus oj
indicates natural frequencies in the frequency regions 20 < < 30, 130 < a>2 < 140,
and 340 < < 350. Further calculations were carried out in each of these regions
with a much smaller frequency step. Because the lumped-mass model of only three
masses could hardly give reliable results for the third mode, only the first two modes
were recomputed; these were found to be a>, = 25.03 and (O2 = 138.98. The mode
shape at co2 is plotted in Fig. 12.6-3.

12.7 COUPLED FLEXURE-TORSION VIBRATION

Natural modes of vibration of airplane wings and other beam structures are often
coupled flexure-torsion vibration, which for higher modes differ considerably from
those of uncoupled modes. To treat such problems, we must model the beam as
shown in Fig. 12.7-1. The elastic axis of the beam about which the torsional
rotation takes place is assumed to be initially straight. It is able to twist, but its
bending displacement is restricted to the vertical plane. The principal axes of
bending for all cross sections are parallel in the undeformed state. Masses are
lumped at each station with its center of gravity at distance from the elastic axis
and 7, is the mass moment of inertia of the section about the elastic axis, i.e.,
J, = + m¡cj.

7/. nil