Page 372 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.4 Mode-Acceleration Method 359

We note here that if F{a,t) and M(a,t) were static loads, the last term
containing the acceleration would be zero. Thus, the terms
^ <pXa)<p,(x)

(11.4-3)

must represent influence functions, where a(a, x) and f3(a, x) are the deflections
at X due to a unit load and unit moment at a , respectively. We can, therefore,
rewrite Eq. (11.4-2) as

y{x,t) = F{a,t)a{a, x) F M{a, t)[3{a, x) - ^ 4.4^
iO,
Because of coj in the denominator of the terms summed, the convergence is
improved over the mode-summation method.
In the forced-vibration problem in which F(a, t) and M{a, t) are excitations,
Eq. (11.3-4) is first solved for q^{t) in the conventional manner, and then substi­
tuted into Eq. (11.4-4) for the deflection. For the normal modes of constrained
structures, F{a,t) and M(a,t) are again the forces and moments exerted by the
constraints, and the problem is treated in a manner similar to those of Sec. 11.3.
However, because of the improved convergence, fewer number of modes will be
found to be necessary.
Example 11.4-1
Using the mode-acceleration method, solve the problem of Fig. 11.3-2 of a concen­
trated mass attached to the structure.
Solution: By assuming harmonic oscillations,
F(a,t) = F(a)e'^‘
q,{t) =q,e‘^‘

y{x,t) =y(x)e""'
By substituting these equations into Eq. (11.4-4) and letting x = a,

y{a) = F{a)a{a,a) + o) 2^-----^—

Because the force exerted by mo on the structure is
F{a) = m„a)-y{a)