Page 368 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.3 Normal Modes of Constrained Structures 355

Figure 11.3-2.
to the natural frequencies of the constrained modes, and the mode shapes of the
constrained structure are found by substituting the Qj into Eq. (11.3-1).
If, instead of springs, a mass is placed at a point jc = a, as shown in Fig.
11.3-2, the force exerted by on the beam is
F{a,t) = -mf^y{a,t) = (11.3-9)
i
Thus, in place of Eq. (11.3-8), we obtain the equation

<7, = \o}^mQip,{a)Y.Qj^j{a) (11.3-10)

Example 11.3-1
Give a single-mode approximation for the natural frequency of a simply supported
beam when a mass is attached to it at jc = //3.
Solution: When only a single mode is used, Eq. (11.3-10) reduces to
- it>^) = o)^mQ(p\{a)
Solving for we obtain

( ^ ) ^ 1 + ^< p^(a)
M l
For the first mode of the unconstrained beam, we have
E l r- . TTX
O»! = 7T ^2 <p,ix) = ^/2sm —

<p,| j j = ^|2sm y = X 0.866
= M = mass of the beam
Thus, its substitution into the preceding equation gives the value for the one-mode
approximation for the constrained beam of
V 2

(i) ■ i +iW
M
Example 11.3-2
A missile is constrained in a test stand by linear and torsional springs, as shown in
Fig. 11.3-3. Formulate the inverse problem of determining its free-free modes from
the normal modes of the constrained missile, which are designated as and il,.