Page 366 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.3 Normal Modes of Constrained Structures 353

Multiplying these two equations by dx and ijj, dx, adding, and integrating, we
obtain

HQj (A,) dx + dx
j •'o 7

= ( { x , t)(pj dx + i Tl{x, dx (11.2-7)
p
A) ''()
If the q's in these equations are generalized coordinates, they must be
independent coordinates that satisfy the equation

¿i, + = -^[^j'^p{x,t)(p,dx + j^Ti(x,t)il>^ dx^ ( 11.2-8)
We see then that this requirement is satisfied only if

rl ( 0 i f y ^ i
/ {nup:(f, + 7iA,iA,) dx (11.2-9)
d(\ M, if y = i
which defines the orthogonality for the beam, including rotary inertia and shear
deformation.

11.3 NORMAL MODES OF CONSTRAINED STRUCTURES
When a structure is altered by the addition of a mass or a spring, we refer to it as a
constrained structure. For example, a spring tends to act as a constraint on the
motion of the structure at the point of its application, and possibly increases the
natural frequencies of the system. An added mass, on the other hand, can decrease
the natural frequencies of the system. Such problems can be formulated in terms
of generalized coordinates and the mode-summation technique.
Consider the forced vibration of any one-dimensional structure (i.e., the
points on the structure defined by one coordinate x) excited by a force per unit
length /(x, t) and moment per unit length M(x, t). If we know the normal modes
of the structure, and (Piix), its deflection at any point x can be represented by
(11.3-1)

where the generalized coordinate must satisfy the equation
1
j
< 7,(0 + < < 7 ,( 0 = jf{x,t)(p^{x) dx + M(x,t)(p'Xx) dx (11.3-2)
M,.
The right side of this equation is 1 /M- times the generalized force 2,, which can
be determined from the virtual work of the applied loads as = 8W/8q^.
If, instead of distributed loads, we have a concentrated force F(a,t) and a
concentrated moment M(a, t) at some point x = a, the generalized force for such