Page 360 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.1 Mode-Summation Method 347

Consider, for example, the general motion of a beam loaded by a distributed
force p{x, t), whose equation of motion is
[EIy"{x,t)]" + m{x)y{x, t) = {x,t) (H-1-1)
p
The normal modes of such a beam must satisfy the equation
(EIcl>';y - = 0 (11.1-2)
and its boundary conditions. The normal modes are also orthogonal func­
tions satisfying the relation
ri f 0 for j ^ /
jm(x)<f>.4>jdx=\j^^ fory = / (11.1-3)

By representing the solution to the general problem in terms of (f)^(x)

y i x , t ) = 'Z<t>i{x)qi(t) (11-1-4)
i
the generalized coordinate q^{t) can be determined from Lagrange’s equation by
first establishing the kinetic and potential energies.
Recognizing the orthogonality relation, Eq. (11.1-3), the kinetic energy is

T = y y ^ ( x , t ) m ( x ) d x = ' L < i , q j f 4 > i ( t > j > n ( x ) d x

= (11.1-5)
i
where the generalized mass M- is defined as

M-= ^(f)f(x)m(x)dx (11.1-6)
f
•'o
Similarly, the potential energy is

!
i/ = y f ‘EIy"^(x, i) d*: = 4 E 'Edidj dx
^ •'o ^ / j •'o
= J i (11.1-7)
i
where the generalized stiffness is

Kt= i ‘El[d)'l{x)Y dx ( 11.1-8)