Page 311 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

298 Vibration of Continuous Systems Chap. 9

9-31 A simply supported beam has an overhang of length I2, as shown in Fig. P9-31. If the
end of the overhang is free, show that boundary conditions require the deflection
equation for each span to be
/ sin pL \
= C ( s in ^ ^ -

r / cos PI2 + cosh PI2
<t>2 = A COS fix + cosh fix - I sin ¡31^ + sinh j8/-

where x is measured from the left and right ends.

7/A^/
t: -h~ Figure P9-31.

9-32 When shear and rotary inertia are included, show that the differential equation of the
beam can be expressed by the first-order matrix equation
( ] 1
0 0 0 •A
El
d -1
dx ' y > = 1 0 0 kAG < y
M 0 0 1 M
. V j 0 co^m 0 0 .y)
9-33 Set up the difference equations for the torsional system shown in Fig. P9-33. Deter­
mine the boundary equations and solve for the natural frequencies.

J
K

/V Figure P9-33.

9-34 Set up the difference equations for N equal masses on a string with tension 7, as
shown in Fig. P9-34. Determine the boundary equations and the natural frequencies.

m 7
-o— o— o— o— o-
Figure P9-34.

9-35 Write the difference equations for the spring-mass system shown in Fig. P9-35 and find
the natural frequencies of the system.