Page 364 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 364
Sec. 11.2 Beam Orthogonality Including Rotary Inertia and Shear Deformation 351
Thus, the equation for the aeeeleration of any point x is found as
F{l)<p,(0)<p,(.x) <p,(Q)^,(.x)co, r,
= L M, M, Í 'f (Í) sin oj,(t ~ ^)di
Example 11.1-3
Determine the response of a eantilever beam when its base is given a motion
normal to the beam axis, as shown in Fig. 11.1-3.
Figure 11.1-3.
Solution: The differential equation for the beam with base motion is
[EIy"{x,t)~\' + m{x)[yi,(i) + y(jr, r)] = 0
whieh ean be rearranged to
[Ely"(x,t)]" + m {x )^x,l) = -m{x)y^{t)
Thus, instead of the foree per unit length Fix, t), we have the inertial force per unit
length —m(x)yf^it). By assuming the solution in the form
y(x,t) =
the equation for the generalized coordinate becomes
1 ri
di + / <Pi(x)dx
The solution for then differs from that of a simple oscillator only by the factor
- 1/M- i^ipj{x)dx so that for the initial conditions y(0) = y(0) = 0:
d,iO =
'
■o
11.2 BEAM ORTHOGONALITY INCLUDING ROTARY INERTIA
AND SHEAR DEFORMATION
The equations for the beam, including rotary inertia and shear deformation, were
derived in Sec. 9.6. For such beams, the orthogonality is no longer expressed by
